Calculus Volume by Integration Worksheets

8/20/2019 Calculus Volume by Integration Worksheets

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AP CALCULUS BC

MR. CAMPOS

LAP 10

DIFFERENTIAL EQUATIONS, SLOPE FIELDS, EULER’S METHOD,AREA BETWEEN CURVES, VOLUME, AND ARC LENGTH

Objectives:1. Students will solve differential equations.2. Students will solve applications involving differential equations.3. Students will draw slope fields and solution curves.4. Students will estimate the solution to a differential equation by using Euler’s method. 5. Students will find the area between two curves.6. Students will find the volume of a solid with known cross sections.7. Students will find the volume of a solid of revolution by the disc method.8. Students will find the volume of a solid of revolution by the shell method.9. Students will find the length of an arc of a curve.

Monday Tuesday Wednesday Thursday Friday Jan. 26

7.1 Area betweenTwo Curves

HW #1 – Worksheet

Jan. 27

7.2 Volume by CrossSections

HW #2 – Worksheet

Formula Quiz

Jan. 28

7.2 Volume by Discs

HW #3 – P.465: 5, 9, 11, 13, 17,and 21. Work 5, 9, and 11out without calculator.On the rest, set up andthen evaluate on yourcalculator.

Jan. 29

7.2 Volume by DiscsContinued

HW #4 – Worksheet

Jan. 30

7.3 Volume by Shells

HW #5 – P.474: 5,9,15,17,25,29,

30,45

Work 5,9,15 out.Just set up 17, 25, 29, 30.Use calc on 45.

Feb. 2

7.1 – 7.3Area and Volume

HW# 6 - Worksheet

Feb. 3

7.4 Arc Length, Area,and Volume

HW #7-Worksheet

HW #8 -Make a review

Feb. 4

TEST7.1-7.4

Look at the following Geogebra animations for volume by cross sections, disks/washers, and shells:

For cross sections:http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htm

For disks/washers:http://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htm

For shells:http://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_3_7/shell.htm

http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htmhttp://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_3_7/shell.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_3_7/shell.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_3_7/shell.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htmhttp://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htm


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7.1 Area of a Region Between Two Curves

To find the area bounded by two functions

and y f x y g x on the interval [ a , b]:

Area =

To find the area bounded by two functions

and x f y x g y on the interval [ a , b]:

Area =

______________________________________________________________________________________Ex. Find the area bounded by the following graphs. Draw a figure, and shade the bounded region.

(a) 23 and 1 y x y x

(b) 25 and 1 x y x y


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(c) 6 and 2 f x x g x x . Use your calculator.

______________________________________________________________________________________Ex. Set up the integrals needed to find the area of the shaded region.

f x

g x

Homework : Worksheet


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7.2 Volumes of Solids with Known Cross Sections

Today we will find the volume of a solid whose cross sections are familiar geometric shapes, such assquares, rectangles, triangles, and semicircles.

For cross sections of area A x taken perpendicular to the x-axis, Volume =

For cross sections of area A y taken perpendicular to the y-axis, Volume =

There are some great Calculus applets using Geogebra that can help you see these solids. Unfortunately youcannot see them on your tablet because it can’t do Java, bu t you can look at them on your computer at home.

I especially like the ones on Paul Se eburger’s Dynamic Calculus Site, at http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htm

http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htmhttp://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htmhttp://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htm


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Ex. Set up the integrals needed to find the volume of the solid whose base is the area bounded by

the lines 2 and 2 3 y x y x and whose cross sections perpendicular to the x-axis are thefollowing shapes.

(a) Rectangles of height 4

(b) Semicircles

_______________________________________________________________________________________Ex. Set up the integrals needed to find the volume of the solid whose base is the area bounded by the

circle 2 2 9 x y and whose cross sections perpendicular to the x-axis are equilateral triangles.

(Area of an equilateral triangle =2 3

4

s

, where s = side of a triangle)

_______________________________________________________________________________________

Ex. The base of a solid is bounded by 2 , 0, and 2. y x y x For this solid, each cross section perpendicular to the y-axis is square. Set up the integral needed to find the volume ofthis solid.

Homework : Worksheet


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7.2 Volume: The Disc MethodIf we revolve a figure about a line, a solid of revolution is formed. The line is called the axis of revolution.The simplest such solid is a right circular cylinder or disc.

To find the volume of thesolid, we partition it intorectangles, which arerevolved about the axis ofrevolution.

Each disc is a thin cylinder standing on its side.Volume of cylinder =Volume of disc =

Adding the volumes of all of the discs together,Volume of solid

To get the exact volume,Volume of solid =

Volume about horizontal axis by discs: V =

Volume about vertical axis by discs: V =

The disc method can be extended to cover solids of revolution with a hole in them.This is called the washer method.

If

R x is the outer radius and

r x is the inner radius:

Volume about horizontal axis by washers: V =

Volume about vertical axis by washers: V =

Things to remember :In the disc or washer method:1) The representative rectangle is always perpendicular to the axis of revolution.2) If the representative rectangle is vertical, you will work in x's.

If the representative rectangle is horizontal, you will work in y's.

See the Calculus applet on volume by discs and washers at Paul Se eburger’s Dynamic Calculus Site, at http://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htm

http://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_2_13/washer_ex4.htm


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Ex. Find the volume of the solid formed by revolving the region bounded by the graphs of the givenequations about the indicated axis.

29 , 0, 0 y x x y (a) about the x-axis (b) about the line 2 y

_______________________________________________________________________________________(c) about the y-axis (d) about the line 2 x

_______________________________________________________________________________________Ex. Find the volume of the solid formed by revolving the region bounded by the graphs of

2 22 and y x x y x about the line y = 3.

Homework : P.465: 5, 9, 11, 13, 17, and 21Work 5, 9, and 11 out to find the answer (no calculator)

On the rest of the problems, set up the integral(s) needed and use yourcalculator to evaluate.


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7.3 Volume by the Shell MethodTo find the volume as y f x is revolved aroundthe y-axis by discs, we sliced a horizontal rectangleand looked at the cylinder formed.

To find the volume of this same solid by the shellmethod, we look at a vertical rectangle revolvedaround the y-axis. It would form a hollowed-outcylinder or shell.

If we flattened out the shell, wewould have a thin rectangularsolid.

Length of the solid =

Height of the solid =

Width of the solid =

Volume of the solid =

Volume about a vertical axis by shells: V =

Volume about a horizontal axis by shells: V =

The radius, r x , is the distance between the rectangle and the axis of revolution.

The height, h x , is the height or length of the rectangle.

With the disc method, the representative rectangle is always perpendicular to the axis of revolution.With the shell method, the representative rectangle is always parallel to the axis of revolution.

It is usually easiest to use discs about a horizontal axis and shells about a vertical axis. If you dothis, you will work in x's. If you use discs about a vertical axis or shells about a horizontal axis,you will work in y's.

See the Calculus applet on volume by discs and washers at Paul Se eburger’s Dynamic Calculus Site, at http://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_3_7/shell.htm

http://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_3_7/shell.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_3_7/shell.htmhttp://higheredbcs.wiley.com/legacy/college/anton/0470183454/applets/ch6/figure6_3_7/shell.htm


9/18

Ex. Set up an integral expression that can be used to find the volume when the region bounded by 2 1 and 0 y x y is rotated about the given axis over the interval [0, 2] by using

shells.(a) y-axis (b) 1 x

_____________________________________________________________________________________Ex. Set up an integral expression that can be used to f ind the volume when the region bounded

by , 0, and 4 y x y x about the given axis by using shells.(a) x = 7 (b) y = 5

_____________________________________________________________________________________Ex. Find the volume of the solid generated by revolving the region between

12 3 and y x y x about the y-axis. Use: (a) shells, (b) discs.(a) (b)

Homework : P.474: 5, 9, 15, 17, 25, 29, 30, 45Work 5,9,15 out (no calculator).Just draw and set up the integrals on 17,25,29,30. Do not evaluate.Use your calculator on 45.


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7.4 Arc Length

Arc length of f x from x = a to x = b:

2

1b

a s f x dx

Arc length of f y from y = c to y = d :

2

1

d

c s f y dy

Ex. Find the arc length of the graph of the given function over the indicated interval.

(a) 3

2 1 0, 4 y x

(b) 2

33 10 8, 27 y x

Homework: Worksheet


11/18

y

CALCULUS BCWORKSHEET ON 7.1

Work the following on notebook paper .Find the area bounded by the given curves. Draw and label a figure for each problem, and show all work. Do notuse your calculator on problems 1 - 3 .

1. 2 2 1, 3 3 f x x x g x x

2. 3 2 23 3 , y x x x y x (see figure on the right)

3. 2 , 2 x y x y Figure for prob. 2 ____________________________________________________________________________________________Find the area bounded by the given curves. Draw and label a figure for each problem, set up the integral(s) needed,and then evaluate on your calculator.

4. 4 , 3 4 f x x g x x

5.

2

, 2 x

f x x g x (see figure on the right)

6. 2ln 1 , cos y x y x Figure for prob. 5

____________________________________________________________________________________________7. (No calculator)Let R be the region in the first quadrant bounded by the (4, 2)

x-axis and the graphs of y x and 6 y x as shownin the figure on the right.(a) Find the area of R by working in x’s.

(b) Find the area of R by working in y’s .

(c) When you found your answers to (a) and (b), was it less work to work in terms of x or in terms of y? ____________________________________________________________________________________________8. (Calculator)Let R be the region bounded by the graphs of

12 , 4 , and , x x y y y

x as shown in the figure

on the right. Find the area of R

R

R


12/18

CALCULUS BCWORKSHEET ON VOLUME BY CROSS SECTIONS

Work the following problems on notebook paper . For each problem, draw a figure, set up an integral, and thenevaluate on your calculator. Give decimal answers correct to three decimal places.

1. Find the volume of the solid whose base is bounded by the graphs of 1 y x and 2 1 y x , with theindicated cross sections taken perpendicular to the x-axis.

(a) Squares (b) Rectangles of height 1

(c) Semiellipses of height 2 (The area of an ellipse is given by the formula A ab , where a and bare the distances from the center to the ellipse to the endpoints of the axes of the ellipse.)(d) Equilateral triangles

2. Find the volume of the solid whose base is bounded by the circle 2 2 4 x y with the indicated crosssections taken perpendicular to the x-axis.

(a) Squares (b) Equilateral triangles(c) Semicircles(d) Isosceles right triangles with the hypotenuse as the base of the solid

3. The base of a solid is bounded by 3 , 0, and 1. y x y x Find the volume of the solid for each of the

following cross sections taken perpendicular to the y -axis.(a) Squares (b) Semicircles(c) Equilateral triangles(d) Semiellipses whose heights are twice the lengths of their bases


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CALCULUSWORKSHEET ON 7.1 – 7.2

Work the following on notebook paper .On problems 1 – 2, use your calculator, and give your answers correct to three decimal places. Each answer musthave supporting work.1. (2004 Form B)

Let R be the region enclosed by the graph of 1 y x , the vertical line x = 10, and the x-axis.

(a) Find the area of R.(b) Find the volume of the solid generated when R is revolved about the horizontal line y = 3.(c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are

equilateral triangles. Find the volume of this solid.

2. (2003) Let R be the shaded region bounded by the graphs of 3 and x y x y e and the vertical line x = 1,as shown.

(a) Find the area of R.(b) Find the volume of the solid generated when R is revolved about the

horizontal line y = 1.(c) The region R is the base of a solid. For this solid, each cross section

perpendicular to the x-axis is a rectangle whose height is 5 times the lengthof its base in region R. Find the volume of this solid.

_________________________________________________________________________________________Do not use your calculator on problems 3 – 4.3. Let R be the region enclosed by the graphs of 2 and 12 . y x y x (a) Find the area of R.(b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares.

Write, but do not evaluate, an integral expression that gives the volume of this solid.(c) The region R is revolved about the x-axis. Write, but do not evaluate, an integral expression that gives the

volume of the solid formed.

4. Let R be the region enclosed by the graphs of 2 41 , , and . y x y y x

(a) Find the area of R.(b) The region R is the base of a solid. For this solid, the cross sections R

perpendicular to the y -axis are semicircles. Write, but do not evaluate,an integral expression that gives the volume of this solid.

(c) The region R is revolved about the vertical line .3 x Write, but do notevaluate, an integral expression that gives the volume of the solid formed.


14/18

CALCULUSWORKSHEET ON AREA AND VOLUME

Work the following on notebook paper. Do not use your calculator.1. Let R be the region bounded by the graphs of 24 and 2. y x y x (a) Find the area of R.(b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares.

Write, but do not evaluate, an integral expression for the volume of this solid.(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotatedabout the horizontal line y = 6.

(d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotatedabout the vertical line 3. x

2. Let R be the region in the first quadrant bounded by the graphs

of 2 , y x the horizontal line y = 6, and the y-axis, as shownin the figure on the right.

(a) Find the area of R.(b) Write, but do not evaluate, an integral expression for the volume

of the solid generated when R is rotated about the vertical line x = 12.

(c) Write, but do not evaluate, an integral expression for the volumeof the solid generated when R is rotated about the horizontal line y = 7.

(d) Region R is the base of a solid. For each y, where 0 6, y the cross section of the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R.Write, but do not evaluate, an integral expression that gives the volume of the solid.

3. Let R be the region bounded by the x-axis, the graph of y x , and the line x = 4.(a) Find the area of the region R.(b) Find the value of h such that the vertical line x = h divides the region R into two regions of equal area.(c) Find the volume of the solid generated when R is revolved about the x-axis.(d) The vertical line x = k divides the region R into two regions such that when these two regions are revolved

about the x-axis, they generate solids with equal volumes. Find the value of k .

4. Let R be the region in the first quadrant bounded by the graphs of 21

, , y x y x

the x-axis and the

vertical line x = 3.(a) Find the area of the region R.(b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are

rectangles with height five times the length of the base. Find the volume of this solid.(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

rotated about the horizontal line y = 2.


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CALCULUS BCWORKSHEET ON ARC LENGTH AND REVIEW

Work the following on notebook paper .On problems 1 – 3, find the arc length of the graph of the function over the indicated interval. Do not use yourcalculator on problems 1 - 3.

1. 3

22 1, 0, 13

y x 2. 2

33 , 1, 82

y x 3. 3

ln sin , ,4 4

y x

__________________________________________________________________________________________On problems 4 – 7, sketch the graph of the given function, set up an integral that represents the arc length of thegraph of the function over the indicated interval, and evaluate it on your calculator. Give your answers correct tothree decimal places.

4. 24 , 0 2 y x x 6. sin , 0 y x x

5.1

, 1 3 y x x 7. , 0 2 y x e y

__________________________________________________________________________________________On problems 8 – 9, sketch the graphs of the given functions, and find the area bounded by the graphs of thefunctions. Do not use your calculator.

8. 2 4 1, 1 f x x x g x x 9. 2 , 6 f y y g y y __________________________________________________________________________________________10. (Modified version of 2009 AB 4) ( No calculator )Let R be the region in the first quadrant enclosed by the graphs of 22 and . y x y x (a) Find the area of R.(b) The region R is the base of a solid. For this solid, at each x, the cross section perpendicular to the x-axis

has area sin .2

A x x

Find the volume of the solid.

(c) Another solid has the same base R. For this solid, the cross sections perpendicular to the y-axis are squares.Write, but do not evaluate, an integral expression for the volume of the solid.

(d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotatedabout the horizontal line 3. y

(e) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotatedabout the vertical line 3. x

11. (2001 AB 1) ( Use your calculator .)Let R and S be the regions in the first quadrant shownin the figure. The region R is bounded by the x-axisand the graphs of 32 y x and tan y x . The region

S is bounded by the y-axis and the graphs of 32 y x and tan y x .

(a) Find the area of R.(b) Find the area of S .(c) Find the volume of the solid generated when S

is revolved about the x-axis.(d) The region S is the base of a solid. For this solid,

each cross section perpendicular to the x-axis isa semicircle. Find the volume of this solid.


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01 3456789595489

01 3456789595489

01 34565

01 3456501 34565

01 34565

HW –P. : , , 11, 1 , 1 , an 1.

Work 5, 9, and 11 out without calculator.On the rest, set up and then evaluate on your calculator.


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01 3456789595 81 5 665 81

01 3456789595 81 5 665 81

01 3455 98 5101 345955 9

01 345

HW 5 – P.474: 5,9,15,17,25,29, 30,45

Work 5,9,15 out. Just set up 17, 25, 29, 30. Use calc on 45.


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Calculus Volume by Integration Worksheets

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