ITESO: Integral Calculus, Student Learning Guide

Guía de Aprendizaje Cálculo Integral DMAF ITESO

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ITESO DEPARTMENT OF MATHEMATICS AND PHYSICS

BASIC ACADEMIC UNIT: CALCULUS

STUDENT LEARNING GUIDE

COURSE: INTEGRAL CALCULUS

INDEX Competency to develop Expected products Area under the curve by sums, the definite integral Knowledge construction Knowledge consolidation Examples Area, volume and arc length through integration Knowledge construction Knowledge consolidation Examples Series Knowledge construction Examples Knowledge consolidation Examples Additional exercises

PRE-REQUISIT: To have coursed and passed Differential Calculus. COURSE DESCRIPTION: Integral Calculus is one of three calculus courses that Engineer students must course in ITESO. The other two courses are Differential Calculus (before Integral Calculus) and Advanced Calculus (or Multivariate Calculus or Differential Equations, depending on the career), after this course. These three courses are situated in the area that refers to “Professional Knowledge” in ITESO´s engineering careers. COMPETENCY TO DEVELOP IN THE COURSE: “The student solves accumulation problems in engineering contexts to model and determine total results of the effects of change through the application of the concepts of `integral´ and `series´, based on the criteria of pertinence, argumentation, resource management and technique”. SPECIFIC COMPETENCIES:

In the framework of an applied situation, the student:


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a) Identifies and constructs the mathematical model through which the situation can be solved (the modelling process is understood as the construction of new models or the adaptation of models that are already established).

b) Operates the constructed model (including the selection of appropriate instruments to carry out the operations and the execution of those operations).

c) Establishes conclusions about the behavior of the addressed situation and expresses these conclusions in oral and written form.

EXPECTED FINAL PRODUCTS AND THEIR EVALUATION: PRODUCTS:

o Four department exams and one final exam. o Integration activities (projects, team presentations, portfolios, applied problems, etc.). o Homework and readings for each theme.

PRODUCT EVALUATION

The components of the course´s program that are included in the final grade are the following: (a) Written exams (b) Homework and/or readings (c) Integration activities (projects, team presentations, portfolios, applied problems, etc.) The points assigned to each product are:

Department Exams (4) and Final Exam

80 points

Homework and readings 10 points

Integration activities 10 points

o There are four department exams, and the study themes for each are:

o Department Exam I: Accumulations, Riemann Sums, Definite Integral. o Department Exam II: Integration Techniques, Improper Integrals and Integration by

Approximation. o Department Exam III: Applications of Integrals o Department Exam IV: Sequences, Numerical Series and Power Series.

The final exam includes study themes from the entire course. o In order to pass the course, it is necessary (but not sufficient) for the student to accumulate a

minimum grade point average of 50% in the exams. TEXTBOOKS AND CALCULATOR:

Calculus

– James Stewart

– Seventh edition

– Cengage Learning Calculus Early Transcendentals.

– James Stewart


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– Seventh edition

– Cengage Learning

It is required for students to have a scientific calculator, at the least. COURSE COMPETENCIES: In the following section there is a list of important aspects that contribute to develop the course´s competencies: BEGINNING

AREA UNDER THE CURVE BY SUMS, THE DEFINITE INTEGRAL

1. DEVELOPMENT OF THE COMPETENCY: The student manages applied situations that conduct to the resolution and analysis of problems concerning phenomena of accumulation of variations.

PARTIAL ACHIEVEMENTS: The student:

a) Identifies and constructs mathematical models that represent situations where there is an emphasis on the importance of the total results (or effects) of the processes of change or variation.

b) Recognizes the relationship between phenomena of variation and accumulated results of rates of change using the antiderivative and the integral.

c) Generates the total function of the accumulation of the process of change using the appropriate integration method.

d) Operates the function of accumulation of rates of change or variations in order to solve formulated situations.

e) Arrives at conclusions and expresses them in oral and written form.

NECESSARY INFORMATION TO REACH THESE ACHIEVEMENTS The student:

Determines areas that represent total effects of change, parting from rates of change: Calculates the effect or total result at the end of a given period parting from average rates of change (variations) defined in small time intervals. For example, calculates the total distance travelled by a vehicle, given its instantaneous velocity registered in n second intervals.

Knows and applies the Sigma notation, properties of summations and formulas to add the first n square integers.

Calculates limits of infinite summations: Riemann Sums. Calculates the definite integral of a function using the limit of Riemann sums and visualizes the definite integral as the area between curves.

Definite Integrals and Antiderivatives: Recognizes the relationship between the derivative and the integral (Fundamental Theorem of Calculus). Relates it to phenomena of total results parting from variations.

Knows and applies the net change theorem in situations that involve variation phenomena.

Interprets the definite integral as a tool used to calculate the average value of a function at a given interval.

BEGINNING

LEARNING SITUATIONS TO REACH THESE ACHIEVEMENTS


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1. KNOWLEDGE CONSTRUCTION FOR THE AREA UNDER A CURVE AND THE DEFINITE INTEGRAL

WHAT THE STUDENT DOES

Analyzes and interprets the primitive function as the antiderivative of a function, and Integral Calculus as the inverse process of Differential Calculus in problems that refer to exact sciences (area under the curve), natural sciences (exponential growth) and economy (supply and demand).

Manifests his/her learning by solving proposed exercises and problems.

Analyzes and identifies the best way to solve an indefinite integral.

Collaboratively and individually solves exercises about indefinite integrals.

Reads and analyzes how to calculate the area between a curve and the “x” axis (abscissa).

Estimates the area under a curve through approximations with right/left rectangles or midpoints.

Applies the necessary practical criteria to adequately formulate problems that involve the area under a curve with lines and with quadratic functions, through Riemann Sums.

Collaboratively and individually solves exercises about the area under the curve.

Investigates and recognizes the importance of Riemann Sums and their contribution to the development of Integral Calculus.

BEGINNING 2. KNOWLEDGE CONSOLIDATION FOR THE AREA UNDER A CURVE AND THE DEFINITE INTEGRAL


Analyzes and discusses the importance of Riemann Sums and the Mean Value Theorem in the process of calculating an area or of evaluating a definite integral.

Solves problems using procedures established in integral calculus, with the help of the professor.

Presents written reports that contain the solutions of problems, with the support of algebraic procedures and graphic visualization of areas under curves.

Analyzes the pertinence and utility of Riemann Sums when finding the area under linear and quadratic functions.

BEGINNING EXAMPLES OF SITUATIONS THAT HELP GENERATE THIS KNOWLEDGE: EXAMPLE 1: In Spain´s 2008 F1 Grand Prix, Fernando Alonso and Kimi Raikkonen´s automobiles are next to each other at the beginning of the race. The speed of Kimi Raikkonen´s automobile is represented by (VR), and the speed of Fernando Alonso´s automoblie, by (VA). In the following table, you can observe each vehicle´s speed during the first 10 seconds of the competition:

How many meters does Fernando Alonso´s automobile overtake Kimi Raikkonen´s after the first 10 seconds?


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Procedure:

Identify the distance formula. tvD Plot the values given in the table with the correct conversion of velocity in m/s.

Establish the distance formula as the formula used to find the area of a rectangle. vthbA

Obtain the area of every rectangle in each 1 second time interval with right extreme rectangles and left extreme rectangles. Distance with right extreme rectangles:

Distance with left extreme rectangles:

Interpret the results: The distance that Alonso´s automobile overtakes Raikkonen´s (with right extremes) is 59 m. The distance that Alonso´s automobile overtakes Raikkonen´s (with left extremes) is 53 m.

Therefore, Alonso´s advantage is 5953 D , which represents the area under the curve. EXAMPLE 2

The administration of a drinking water station must find a balance between the consumption of drinking water and its production so that it can assure an adequate supply for its population. To reach this balance between supply and demand, instant consumption is registered through special sensors that continuously measure the water flow. Based on these measurements, it is possible to calculate mean values of consumption by the hour and to represent them in a table:

time (hr) 3m / min time (hr) 3m / min

7 1490 19 3500

8 1510 20 2600

9 1100 21 -500


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10 1600 22 -600

11 1600 23 -2500

12 1100 24 -3000

13 1200 1 -3500

14 1600 2 -3500

15 600 3 -3000

16 700 4 -2800

17 1600 5 -2000

18 2000 6 -200

Table 1: drinking water mean

consumption rate by the hour. With the previous information, and after plotting the information in the table, calculate:

1. The total consumption in 24 hours. 2. How much was consumed from 7 to 16 hours? 3. In which moment is the accumulated consumption at its maximum? 4. Is there a balance between what is consumed and what is produced? 5. In what moment is the water consumed more quickly?

EXAMPLE 3: Tired of being bumblebee victims, honey bees learned to organize collectively and defend themselves when faced with these predators. Presently, as soon as the bumblebee penetrates their territory, the honey bees regroup and strangle the insect: they obstruct its breathing, block its nostrils and its respiratory movements. On the other hand, Asian bees regroup around the predator bumblebee and afterward become fierce by producing strong heat until it submits. One honey bee population starts off with 1000 bees, and increases proportionately through time (in

weeks) established by the following expression: ttttf 544010)( 23

How many bees will be part of the hive after 12 weeks? Procedure:

Plot the function ttttf 544010)( 23

Establish the area under the curve as the result of the definite integral

12

0

23 544010 dtttt and the

number of bees after 12 weeks.


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Evaluate the area under the curve or the definite integral with Riemann Sums for n rectangular subintervals (right or left). In this case, right intervals are used.

for Therefore:

By evaluating the integral with Riemann Sums, after 12 weeks the bee population has increased in 78768 bees. BEGINNING

AREA, VOLUME AND ARC LENGTH THROUGH INTEGRATION


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2. DEVELOPMENT OF THE COMPETENCY: The student manages applied situations that conduct to the resolution and analysis of problems concerning phenomena of accumulation (mainly geometric problems): calculation of surface areas limited by curves; calculation of volumes of solids of revolution with the use of definite integrals; calculation of curve lengths.

PARTIAL ACHIEVEMENTS: The student:

Formulates the area of a surface limited by curves.

Calculates the surface area with definite integrals.

Identifies the graphic form of a volume by spinning a closed region around a given axis.

Based on the characteristics of a volume, selects the adequate method to calculate it.

Calculates volumes by operating with functions.

Calculates the area under a curve with integration techniques: by parts, by powers of trigonometric functions, by trigonometric substitution and by partial fractions.

Calculates arc lengths.

Calculates surfaces of revolution.

Approximately calculates integrals with the Trapezoid Method or Simpson´s Rule.

Arrives at conclusions and expresses them in oral and written form.

NECESSARY INFORMATION TO REACH THESE ACHIEVEMENTS Areas:

Identifies the representative rectangle in order to calculate the area of a closed, flat region.

Knows and applies techniques to calculate the area between two curves.

Determines the area between curves, either on flat surfaces under the “x” axis or to the left of the “y” axis.

Formulates and solves the integrals that correspond to the calculation of the area of a flat surface.

Conceptualizes and solves improper integrals.

Volumes and surfaces:

Identifies that the volume of a solid of revolution is generated from making a closed region turn around an axis, either one of the coordinate axes or a straight line parallel to one of those axes.

Deduces and calculates volumes of solids of revolution with the use of one of the following methods:

o Disk integration o Ring integration o Shell integration

Identifies the representative element (disk, ring or shell) to calculate the volume of a solid of revolution.

Formulates and solves the integrals that correspond to the calculation of the volume of solids of revolution.

Formulates and solves the integrals that correspond to the calculation of the area of surfaces of revolution.

Approximate Integration Methods:


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Graphically deduces the approximate calculation of integrals with the use of trapezoids or parabolic curves.

Arc Length:

Identifies the length of a curve at a given interval, formulates the integral that permits the length to be calculated and uses the adequate technique to solve the integral.

BEGINNING


1. KNOWLEDGE CONSTRUCTION FOR CALCULATING AREA, VOLUME AND ARC LENGTH THROUGH INTEGRATION


Constructs the mathematical model from the graph. Identifies the area or the closed region. Turns the closed region around the specified axis and identifies the volume. Calculates the volume.

Identifies the area between two curves, the volume or the arc length and calculates them using definite integrals.

Reads and analyzes information about the calculation of volumes, area between two curves or arc length.

Applies practical criteria necessary for the adequate formulation of problems that involve areas, volumes and arc lengths.

Manifests his/her knowledge after solving proposed exercises and problems.

BEGINNING 2. KNOWLEDGE CONSOLIDATION FOR CALCULATING AREA, VOLUME AND ARC LENGTH THROUGH INTEGRATION


Through independent work (out of class), he/she studies the proposed situations and questions, to later elaborate their solutions.

Solves problems by making use of procedures learned with the guidance of the professor.

Presents the strategies and procedures used to answer the questions in the proposed situations to the rest of the group. Answers questions made by classmates and by the professor.

Individually solves situations similar to those proposed by his/her classmates and by the professor.

Presents written reports that contain solutions to problems, with algebraic procedures and graphic visualization of areas between two curves and of areas that turn to obtain volumes.

BEGINNING EXAMPLE OF A SITUATION THAT HELPS GENERATE THIS KNOWLEDGE: The owner of a restaurant is offered two options for the recipient of a beverage. He´s interested in the one with the greater capacity and he also wants to know the difference between both volumes.

Option 1. Volume generated when the function turns around the “y” axis, with a height

of 18 cm (considering that each unit in the Cartesian plane is 1 cm). Option 2. Volume generated from the region delimited by the following functions:

from ,

x=3 from


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from

And turning around the “y” axis, with a height of 20 cm (considering that each unit in the Cartesian plane is 1 cm). The closed region that is to be turned is presented in Figure 1. a) Draw the solid formed in each case. Include the plane figure that is turned. b) Calculate the volume for each option. c) Establish the best option for the owner of the restaurant and justify your answer.

Figure 1: Region to be turned for Option 2.

BEGINNING

SERIES

3. DEVELOPMENT OF THE COMPETENCY: The student conceptualizes, operates, models and solves problems related to infinite sums and the expansion of functions in power series.

PARTIAL ACHIEVEMENTS: The student: o Correctly obtains the representation of specific problems in the form of geometric series. o Correctly calculates the sum of a geometric series. o Determines the behavior of the partial sums of a series by inspection. o Adequately calculates the sum of a series parting from its partial sums. o Applies the appropriate criteria to determine whether a series converges or diverges. o Uses pertinent methods to obtain the representation of a given function in the form of a

power series. o Correctly applies the criteria and processes to obtain the interval of convergence. o Calculates the derivative and/or integral, term by term, in the obtained series. o Replaces an adequate numerical and/or algebraic value in the series to determine the

required sum.

NECESSARY INFORMATION TO REACH THESE ACHIEVEMENTS The student:

Determines whether a sequence converges or diverges.

Calculates the sum of infinite series, when possible.

Finds the n-th partial sum in a series.

Applies convergence/divergence tests for series, such as: a) Integral test


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b) Divergence c) Comparison test d) Limit comparison test e) Ratio test f) Alternating series test

Determines if a given series is absolutely convergent, conditionally convergent or divergent.

Represents functions through power series around a specific point.

Uses series to calculate approximate values in applied problems.

To reach these achievements, the student:

Has previous knowledge of: o Elemental algebraic operations. o Elemental geometry (areas, perimeters, polygons) o Concepts of sequences and series. o Derivation and integration of elemental functions (particularly polygonal) o Fundamental trigonometric identities

Is capable of: creativity, analysis, organization, planning, team work and independent work. BEGINNING


1. KNOWLEDGE CONSTRUCTION FOR SERIES


Identifies and discusses the fundamental concepts associated to numerical series (geometric, telescopic and others) and to the power series.

Solves exercises and problems individually and/or collaboratively in class and as homework.

Investigates autonomously in literature indicated or recommended by the professor, as well as in other prestigious sources of information.

Applies software associated to the content of the course.

BEGINNING EXAMPLES OF SITUATIONS THAT HELP GENERATE THIS KNOWLEDGE: 1.- An object is situated on vertex A of an equilateral triangle ABC with each side 3 m long. Always advancing in a straight line, it goes to the midpoint D of side AB, then continues until it reaches the midpoint E of side BC, after that it goes to the midpoint F of side AC and finally returns to point D. Now, it begins a similar path to the one it just did, but with the midpoints of the triangle DEF (that is, it goes to the midpoint of side DE, then to the midpoint of side EF, after that to the midpoint of DF and finally returns to the midpoint of side DE). If this process were to continue indefinitely, answer the following:

a) Obtain the representative figure. b) Obtain the series that represents the distance travelled by the object. c) What is the total distance that the object may travel? d) Is the series in (b) convergent or divergent?

2.- Legend says that the inventor of chess: the Brahmin Lahur Sissa, also known as Sissa Ben Dahir, Heard that King Ladava was sad because of the death of his son, so he offered the king the game of chess as


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entertainment so that he would forget his sorrows; the king was so satisfied with the game, that he wanted to thank the young man by granting him whatever he wanted. Sissa wanted only wheat and asked the king for one grain of wheat for the first square of the chess board, double for the second, and so on, doubling the amount of wheat in the previous box for the next box, until he completed the 64 boxes. Ladava agreed to the petition, but when he did the calculations, he realized that the petition was impossible to grant. Let us do some calculations to find out why. a) How many grains of wheat would the king have to give the inventor? b) If a tenth of a gram is assigned to each grain of wheat, how many tons would all of the wheat grains weigh? If the King decided to pay a ton per second, in how many centuries would his debt be covered? c) Write the strategy that you used to answer the questions. What difficulties did you have? How did you overcome them? 3.- A sequence of squares is to be constructed to cover an area of 9 m2 by following this method: the side of every square in the sequence is the segment that unites the midpoints of two consecutive sides of the square that precedes it. If this process were to continue indefinitely, answer the following:

a) Obtain the representative figure. b) Obtain the series that represents the area. c) What size should the initial square be? d) Is the series in (b) convergent or divergent?

4.- Given the telescopic series

212

1

nn

a) Determine the sum of the series. b) Is the series convergent or divergent?

5.- Develop the function 1

( )5 2

f xx

around 2c as a power series, in the following forms:

a) Expressing the results in compact form, with the help of the sigma symbol.

b) Rewriting 1

( )5 2

f xx

as a geometric series centered in 2c .

c) Developing the function 1

( )5 2

f xx

in a Taylor series around 2c .

5.- Using the Maclaurin series of !2

1!6!4!2

1cos)(2

0

642

n

xxxxxxf

n

n

n

,

a) Calculate the value of the integral

1

22

0

cos x dx with an error of less than 410 .

b) Determine for what value of the constant b the series 3 2 4 8 16 ...b b b b converges to 9. BEGINNING

2. KNOWLEDGE CONSOLIDATION FOR SERIES


Continually reinforces the fundamental concepts associated to numerical series (geometric, telescopic and others) and to the power series.


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Solves exercises and problems individually and/or collaboratively in class and as homework.

Participates in class, poses and answers creative questions, questions beyond the apparent, digs deeply and searches for novelty answers.

Applies software associated to the content of the course.

BEGINNING EXAMPLES: 1. Determine whether each of the following series converges or diverges. Determine the result of the

sum for those that converge:

a) 1 1 !n

n

n

b) 2

5

2

6 8n n n

c) 3 2

2 30

2

3

n

nn

2.- A sequence of squares is to be constructed by following this method: the side of every square in the sequence is the segment that unites the midpoints of two consecutive sides of the square that precedes it. What size must the first square be in order for this sequence of squares to cover an area of 9 m2 when each square is placed one after the other? 3.- It is known that certain medication has an average life span of 2 hours in the human body, and that levels of over 500 mg in the body damages the organism. Determine the maximum dosage of this medication that can be administered every 4 hours to a person during a long period of time so that their organism is not exposed to damaging effects.

4.- Expand the function 1

( )2

f xx

in the Taylor series around 1c .

5.- Expand the following functions in the Maclaurin series: a) ( ) cosf x x

b) 1

( ) ln1

xf x

x

6. Given the function 2

( ) xf x xe :

a) Obtain its representation in the Maclaurin power series, detailing at least the first 6 terms. b) Determine the interval of convergence of the series obtained in (a).

c) Deriving the series in (a) term by term, obtain 1

1

( 1) 2 1

!2

n

nn

n

n

.

d) Calculate the integral of

1

0

2

dxxe x using the first 6 terms of the series obtained in (a).

7. Given the geometric series x

xn

n

1

11


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a) Obtain a representation in series for the function x

xf

1

1.

b) With the help of (a), determine a representation in a power series for xxf 1tan .

c) With the use of (a), prove that ...12

11...

7

1

5

1

3

11

4

n

n

d) With the use of (c) and the identity ba

baba

tantan1

tantantan

, prove that

3

1tan

2

1tan

4

11

.

BEGINNING ADDITIONAL CONSOLIDATION EXERCISES

Antiderivatives and Integrals

Calculate the following integrals, through Riemann sums:

a) b) c)

Use Riemann sums to calculate the area under the curve of the following functions:

a) f(x) = x+2; x ϵ [0,4] b) g(x) = x2 +2x +1; x ϵ [0, 3]

Estimate the area under the graph of f(x) = cos(x) from x = 0 to x = by using six rectangles of

approximation. Use the right extreme points of each subinterval to calculate the approximate area. Repeat the same process, but this time with the extreme left points of each subinterval. Are both results the same? A competitor´s speed increased constantly during the first three seconds of a race. The following table shows her speed in half second intervals. Find the superior and inferior estimations for the disance she travelled in these three seconds.

t (seconds) 0 0.5 1 1.5 2 2.5 3 v (meters/second) 0 6.2 10.8 14.9 18.1 19.4 20.2

There was an oil leak that occured with a speed of r(t) liters per hour. The speed decreased with the passing of time. The speed´s values are shown in the table below, in two hour intervals. Calculate inferior and superior estimations for the total amount of oil that leaked.

t(hours) 0 2 4 6 8 10 r(t) (liters/hour) 8.7 7.6 6.8 5.7 5.3

The following graph shows the acceleration of an automobile that partsfrom a motionless state and travels to a speed of 120 km/h, during a period of 30 seconds. Estimate the distance it travelled during this period.


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Given the following graph, estimate using 5 subintervals, first through inferior sums and then

with superior sums.

Prove that:

a) = b) =

If F(x) = where f(t) is the function which is plotted below, ¿which of the following values is the

greatest? a) F(0) b) F(1) c) F(2) d) F(3) e) F(4)

Without evaluating the integrals, prove the following:

a) c)

b) 2 ≤ 2 d)

Calculate the derivative of each expression:

a) b) h(x) = c) h(x) =

Verify that the function 2ln 1

arctan 12

xy x x

, satisfies the differential equation:

2

2

11 " ' ln

1x y xy y

x


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Verify that the function 2arc 9 1

3

xy x sen x

, satisfies the differential equation:

2 29 " ' 9x y xy y x

Are the following results correct?

a) b)

Suppose that you want to measure the area occupied by a window 6 meters wide and 4 meters high, like the one shown below, which has a parabolic form on the superior part:

A particle that was travelling across a straight line was observed uninterruptedly during 9 seconds from a set point P on the line. This allowed a function to be established relating the distance from the particle s to point P (measured in meters) and the observation time t (measured in seconds). In the table below are the accelerations of the particle in the indicated instants. After 2 seconds of observation, the particle was passing through point P, travelling left at a speed of 15 m/s. Determine the position, speed and acceleration of the particle in the instant that it began to be observed and in the instant that the observation finished.

Time after the observation began

Acceleration

1 s -14 m/s2

3 s -2 m/s2

5 s 10 m/s2

7 s 22 m/s2

Certain amount of a radioactive substance was deposited in a laboratory and after three days it was noted that its mass had reduced to 3.7 grams. If one week after this measurement the mass descended to two tenths of a gram, determine:

a) The rhythm (in grams per day) with which the amount of substance was reducing in the instant in which its mass was one gram.

b) The average life span of the radioactive substance. Yesterday (Thursday) a chemist deposited 3 grams of a radioactive substance in a recipient at 10:00 hrs and


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observed that at 20:00 hrs of that same day its mass was reducing at a rate of 50 mg per hour. When he arrived at the laboratory at 8:00 hrs today, he noted that in that moment the mass of the substance was reducing at a rate of 35 mg per hour. What is the amount of radioactive substance that will remain in the recipient today at 22:00 hrs? The flu virus is spreading quickly in a town with a population of 40,000. The illness is extending at a speed that is directly proportional to the product of the number of sick people times the number of healthy people. If at the beginning of the epidemic 120 people were diagnosed with the flu and after a week there were 800 infected with the virus, determine after how much time 10% of the population will be infected if the adequate measures are not taken to contain it. The laboratories that subjected the Shroud of Turin to the carbon-14 test in the year 1988 assured that the fabric was false because its age dates to approximately 1300 DC. Considering this information and that the average life span of carbon-14 is of 5,730 years, determine:

a) What was the percentage of carbon-14 that resulted from the tests made by these laboratories? b) What percentage of carbon-14 should have been detected in these tests to corroborate the belief

that this was the fabric that wrapped Christ´s body after his death? Last Friday, January 21st, at 8 a.m. a person took certain amount of a radioactive substance to a laboratory that he had found in a crater that had been produced by a meteor´s impact many years ago. On Tuesday, January 25th at 8 a.m., the mass of the substance was proven to be of 15.231 grams. A new measurement was made on Friday, January 28th, at the same time, finding then a mass of 9.086 grams. Is it possible to determine the average life span of this substance based on this information?

If f is a continuous function and = 10, find

If f is a continuous function and , find

BEGINNING

Applications of Integrals and Integration Techniques

The vertices of a trapezoid OABC on a coordinate plane are the points 0,0O , 3,0A , 3,5B

and 0,1C , where each unit on the axes represents 1 cm. Calculate the area of the trapezoid.

Calculate the numerical value of the surface area shown in the graph. It is known to be limited above by the

curve represented through the function 3 2 6y x , on the right by the curve represented through the

function

2

4

yx , and below by the line segment represented through the function

3

2 2

xy :

A company that produces honey packs its product in jars that have the form generated when the surface


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shown below turns around the line R (the curve that lies between the two marked points is parabolic):

R

If we assume that each side of the squares formed in the grid represents a length of 1 cm, determine how many jars are necessary to store a production of 140 liters of honey, if all of the jars are filled to their maximum capacity. Calculate the surface area shown in the following graph, which is limited to the left by the “y” axis,

underneath by the curve 4cos

4

xy

sen x

, to the right by the straight line 4x , and on top by the curve

2

22

25

xy

x

:

Calculate the volume of the solid that results from turning the following surface around the “y” axis,

considering that the curved side of the surface is modelled by the function 2

x

y e

:

In the drawing shown below, there is a cardboard rectangle that measures 24 cm wide and 44 cm long. The fish shaped form drawn inside will be cut out, constructed with the help of two parabolas. Calculate the


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amount of cardboard that will be wasted.

Calculate the area of the anvil shaped region that appears below (where both curved sides are parabolas):

From a curve represented by certain function ( )y f x it is known that:

Its derivative adjusts to a model of the following type: 1

'ya bx

Passes through the point 10

1,3

with a slope m=1

Its slope in the point that corresponds to 4x is 1

4

Parting from this information, determine the slope with which the curve crosses the “x” axis.

Calculate the volume of the solid that is generated when the surface limited by the parabola

24

9

xy , the

" "y axis and the straight line 4y turns around the:

" "y axis.

" "x axis.

Straight line 4y .

Straight line 3x .

Using integration, obtain the formula to calculate the volume of a right circular cone that appears in textbooks. Two designs have been presented to a business manager in order to construct an ashtray. The first of them can be made by rotating the surface plotted in Graph 1 around the straight line R (where the curved line is parabolic), whereas the second by rotating the surface plotted in Graph 2 around the straight line T (where


Página 20 de 24

the curved line is also parabolic). The manager hires you to help him decide which of the two designs uses less material, as well as to know how much more material would be used in one ashtray versus the other. What are your answers to this assignment? (NOTE: In both cases, each side of the squares that are formed in the grids represent 1.5 cm)

Graph 1 R

Graph 2

T

Using disk integration, calculate the volume of the solid generated when the region shown below (where the superior part is a parabolic arc) turns around the straight line:

1y

3x

Afterward, verify if the results you found are correct by calculating these same volumes through Shell integration.

Calculate the following integrals:

1)

0

22

1

5

4 3

rdr

r 2)

3 ln

dx

x x

3) tan xdx


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4) 2 21 4

dx

x x 5)

4

31

10

1

tdt

t 6)

1

2

04 9

dx

x

7) sec tan

d

8) cos 3

2 3x

sen x dx 9) 2 xx e dx

10)

2

3

ln xdx

x 11) ln xdx

x 12) 24x x dx

13) 2arcsen x dx 14) lnsen x dx 15)

32

0

sen d

16) 3

2

0

sen d

17) 21 4cos

sen xdx

x 18. cosx xdx

19) 2ln 6x x dx 20)

1

20

2 1

9 4

x

x

21)

2

xdx

x x

22)

3 5

4 2

1 ln ln

ln ln

x xdx

x x x

As a consequence of an accident suffered by a ship tank, an oil stain has been forming on the ocean´s

surface and its area is increasing at a rate of 2

1

2 2 3

t

t t

square kilometers per hour, where t

represents the number of hours that have passed from the moment that the spill was detected. If two hours after that moment the stain already occupied an area of 5 square kilometers, determine its area the moment that it was detected. In a school with 4,000 students a rumor has begun to spread. To analyze phenomena of this type, sociologists assume that the speed with which the number of people find out about a rumor is directly proportional to the product of that amount and the amount of people that haven´t found out about it. If the rumor began to be spread by two people and, after one week, it was known by 400, determine after how much time half of the students will find out about the rumor. A spring has a natural length of 20 cm. If a force of 25 N is required to keep it stretched to a length of 30cm, how much work is required to stretch it from 20 cm to 25 cm? An aquaruim that is 2 m long, 1 m wide and 1 m deep is filled with water. Determine the work needed to pump out half of the water from the aquarium (the density of the water is 1000 kg/m3).

If f is continuous and = 8, prove that f takes the value of 4 at least once in the interval [1,3].

Determine the numbers b that satisfy that the average of f(x) = 2 + 6x – 3x2 in the interval [0, b] is equal to 3. Determine the average value of f in the interval [0,8], if f(x) is:


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A cup of coffee has a temperature of 950C and it takes 30 minutes to cool down to 610 C in a room with a temperature of 200 C. Use Newton´s Law of Cooling to prove that the coffee´s temperature after t minutes is T(t) = 20 + 75e-kt, where k is approximately 0.02 . What is the coffee´s average temperature during the first half hour?

BEGINNING

Sequences and Series

Suppose that a ball is dropped vertically from a height of 8 meters and that each time it bounces it rises vertically, losing 15% of its height. If the ball continues bouncing indefinitely, what is the total distance that it will go over? Consider a triangle ABC, with a right angle in C, where angle BAC measures 40°. Parting from C, trace a line perpendicular to the hypotenuse AB and name the point where this perpendicular line cuts the hypotenuse as D. After this, beginning at D, trace a line perpendicular to the cathetus BC and name the point where this perpendicular line cuts the cathetus as E. Now, beginning at E, trace a line perpendicular to the hypotenuse AB and name the point where this perpendicular line cuts the hypotenuse as F. If you continue this process indefinitely, what is the sum of the lengths of all of these segments that you construct, if the length of the cathetus AC is 5 feet?

A sequence of squares is to be constructed by applying the following method: the side of each square in the sequence is the segment that joins the midpoints of two consecutive sides of the square that precedes it. Find the size that the first square of the sequence should have so that the entire sequence of squares can cover an area of 9 square meters when they are placed one next to the other.

It is known that certain medication has an average life span of 2 hours in the human body, and that levels of over 500 mg in the body damages the organism. Determine the maximum dosage of this medication that can be administered every 4 hours to a person during a long period of time so that their organism is not exposed to damaging effects. Determine whether each of the following series converges or diverges. Determine the result of the sum for those that converge:

a) 1 1 !n

n

n

b)

25

2

6 8n n n

c)

3 2

2 30

2

3

n

nn

Suppose that x is a positive real number less than 1 and consider the series 3 5 7 ....x x x x

a) Calculate the function f(x) that results from adding this series.

b) Calculate5

7f

.


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c) Determine the first partial sum of this series when, as it is evaluated in 5/7, it differs from the result found in (b) in less than one tenth. This indicates that if we “cut off” the power series at the fifth

term, the value obtained when 5

7x differs from the value of the function in less than one

tenth.

d) Redo (b) and (c), but now with3

7x .

Suppose that the functions ( )f x and ( )h x are developed as power series as indicated below:

2

0

( )3

n

nn

nf x x

1

1 2 1( )

2 1 !2

n

nn

n xh x

n

Determine, in each case, the maximum open interval of x values for which these series are valid.

Develop the function 1

( )2

f xx

with the Taylor Series around 1c .

Determine the order of the error committed when calculating cos44º with the first 5 terms of the Taylor

Series for the function ( ) cosf x x around 4

c

Develop the following functions with the Maclaurin Series:

a) ( ) cosf x x b) 1

( ) ln1

xf x

x

c) f(x) = cosh(x)

Develop the following function with the Maclaurin Series 2

1( )

1f x

x

without applying the general

procedure to calculate the coefficients of the expansion.

Develop the function ( ) arctanh x x with the Maclaurin Series, based on the result obtained in the

previous exercise and by using the relationship that exists between the functions ( )h x and ( )f x .

Develop the function 1

( )2

f xx

with the Taylor Series, around 1c .

Calculate the value of the integral

1

2

0

sen xdx

x with an error of less than610

.

Knowing that 2

1

0

1.462651746...x

e dx , determine where the Maclaurin Series that represents the

function 2

( ) xf x e should be “cut off” to approximate the value of the integral with an error of less than

0.000001 .


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Given the function 2

( ) xf x xe :

a) Obtain its representation as a Maclaurin power series, developing at least the first 6 terms. b) Determine the interval of convergence of the series obtained in (b).

c) Deriving the series in (a) term by term, obtain 1

1

( 1) 2 1

!2

n

nn

n

n

.

d) Calculate the integral

1

0

2

dxxe x by using the first 6 terms of the series obtained in (a).

Given the geometric series x

xn

n

1

11

,

a) Obtain a series representation for the function x

xf

1

1.

b) With the information from (a), determine a representation of xxf 1tan as a power series.

c) With the information from (b), prove that ...12

11...

7

1

5

1

3

11

4

n

n

d) With the information from (c) and the identity ba

baba

tantan1

tantantan

, prove

that3

1tan

2

1tan

4

11

.

BEGINNING

ITESO: Integral Calculus, Student Learning Guide

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