MA 123, Chapter 10

MA 123, Chapter 10:

Formulas for integrals: integrals, antiderivatives, and

theFundamental Theorem of Calculus (pp. 207-233,

Gootman)

Chapter’s Goal:

• Understand the statement of the Fundamental

Theorem of Calculus.

• Learn how to compute the antiderivative of

some basic functions.

• Learn how to use the substitution method to

compute the antiderivative of morecomplex

functions.

• Learn how to solve area and distance traveled

problems by means of antiderivatives.

– p. 263/293

Example 1:

Evaluate the indefinite integral

∫

(t3 + 3t2 + 4t+ 9) dt.

– p. 264/293

Example 2:


∫

6√tdt.

– p. 265/293

Example 3:


∫

t3(t+ 2) dt.

– p. 266/293

Example 4:


∫

x2 + 9

x2dx.

– p. 267/293

Example 5:

Find a formula for A(x) =

∫ x

1

(4t+ 2) dt, that is,

evaluate the definite integral of the function

f(t) = 4t+ 2 over the interval [1, x] inside [1, 10]. Find

the values A(5), A(10), A(1). What is the derivative of

A(x) with respect to x?

– p. 268/293

Example 6:

Compute the derivative of F (x) if

F (x) =

∫ x

2

(t4 + t3 + t+ 9) dt.

– p. 269/293

Example 7:

Compute the derivative of g(s) if

g(s) =

∫ s

5

8√u2 + u+ 2

du.

– p. 270/293

Example 8:

Suppose f(x) =

∫ x

1

√t2 − 7t+ 12.25 dt. For which

positive value of x does f ′(x) equal 0?

– p. 271/293

Example 9:

Find the value of x at which

F (x) =

∫ x

3

(t8 + t6 + t4 + t2 + 1) dt takes its minimum on

the interval [3, 100]. The value of x that gives a

minimum of F (x) is .

– p. 272/293

Example 10:

Find the value of x at which G(x) =

∫ x

−5

(|t|+ 2) dt

takes its maximum on the interval [−5, 100]. The value

of x that gives a maximum of G(x) is .

– p. 273/293

Example 11:

Evaluate the integral

∫ 5

0

(t2 + 1)dt.

– p. 274/293

Example 12:


∫ −5

−7

(

1

t

)2

dt.

– p. 275/293

Example 13:


∫ 2

0

ex dx.

– p. 276/293

Example 14:


∫ 2

−2

(t5 + t4 + t3 + t2 + t+ 1) dt.

– p. 277/293

Example 15:


∫ 12

−6

|t| dt.

– p. 278/293

Example 16:


∫ 5

2

(

3u5 +7

u

)

du.

– p. 279/293

Example 17:

Suppose

f(x) =

2x, x ≤ 2;

8

x, x > 2.


∫ 5

0

f(x) dx.

– p. 280/293

Example 18:


∫ x

0

(t+ 9)2 dt.

– p. 281/293

Example 19:


∫ x

0

√3t+ 7 dt.

– p. 282/293

Example 20:


∫ x

0

1

(5t+ 4)2dt.

– p. 283/293

Example 21:


∫ 5

0

√2t+ 1 dt.

– p. 284/293

Example 22:


∫ 1

0

5 e5x+1 dx

– p. 285/293

Example 23:


∫ 3

0

2x

x2 + 1dx.

– p. 286/293

Example 24:

Compute the derivative of F (x) if F (x) =

∫ x2

0

2t dt.

– p. 287/293

Example 25:

A train travels along a track and its velocity (in miles

per hour) is given by v(t) = 76t for the first half hour of

travel. Its velocity is constant and equal to v(t) = 76/2

after the first half hour. Here time t is measured in

hours. How far (in miles) does the train travel in the

second hour of travel?

– p. 288/293

Example 26:

A train travels along a track and its velocity (in miles

per hour) is given by v(t) = 76t for the first half hour of

travel. Its velocity is constant and equal to v(t) = 76/2

after the first half hour. Here time t is measured in

hours. How far (in miles) does the train travel in the first

hour of travel?

– p. 289/293

Example 27:

A rock is dropped from a height of 21 feet. Its velocity

in feet per second at time t after it is dropped is given

by v(t) = −32t where time t is measured in seconds.

How far is the rock from the ground one second after

it is dropped?

– p. 290/293

Example 28:

Suppose an object is thrown down from a cliff with an

initial speed of 5 feet per sec, and its speed in ft/sec

after t seconds is given by s(t) = 10t+ 5. If the object

lands after 7 seconds, how high (in ft) is the cliff?

– p. 291/293

Example 29:

A car is traveling due east. Its velocity (in miles per

hour) at time t hours is given by

v(t) = −2.5t2 + 10t+ 50.

How far did the car travel during the first five hours of

the trip?

– p. 292/293

Example 30:

What is the average of f(x) = x2 over the interval

[0, 6]?

– p. 293/293

MA 123, Chapter 10

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