Deck 12: Integration and Its Applications

Question 1

Find the cost function for the marginal cost $\frac { d C } { d x } = \frac { 1 } { 40 } x ^ { 2 } + 90$ and fixed cost of $\$ 2000$ (for x = 0).

A) $C ( x ) = \frac { 1 } { 80 } x ^ { 3 } + 90 x + 2000$
B) $C ( x ) = \frac { 1 } { 120 } x ^ { 3 } + 2000 x + 90$
C) $C ( x ) = \frac { 1 } { 120 } x ^ { 3 } + 90 x + 2000$
D) $C ( x ) = \frac { 1 } { 80 } x ^ { 4 } + 2000 x + 90$
E) $C ( x ) = \frac { 1 } { 120 } x ^ { 4 } + 90 x + 2000$

Answer 1

$C ( x ) = \frac { 1 } { 120 } x ^ { 3 } + 90 x + 2000$

Question 2

Use algebra to rewrite the integrand; then integrate and simplify. $\int \frac { x + 14 } { \sqrt { x } } d x$

A) $\frac { 2 } { 3 } x \sqrt { x } + 14 \sqrt { x } + C$
B) $\frac { 2 } { 3 } x \sqrt { x } + 28 \sqrt { x } + C$
C) $\frac { 2 } { 3 } x \sqrt { x } + 42 \sqrt { x } + C$
D) $\frac { 1 } { 2 } x ^ { 2 } + 42 \sqrt { x } + C$
E) $\frac { 1 } { 2 } x ^ { 2 } + 28 \sqrt { x } + C$

Answer 2

$\frac { 2 } { 3 } x \sqrt { x } + 28 \sqrt { x } + C$

Question 3

Evaluate the integral $\int 19 x ^ { 7 } d x$ .

A) $133 x ^ { 6 } + C$
B) $152 x ^ { 8 } + C$
C) $\frac { 19 } { 6 } x ^ { 6 } + C$
D) $\frac { 19 } { 8 } x ^ { 8 } + C$
E) $\frac { 19 } { 7 } x ^ { 7 } + C$

Answer 3

$\frac { 19 } { 8 } x ^ { 8 } + C$

Question 4

Find the indefinite integral and check the result by differentiation. $\int \left( 16 x ^ { 3 } - 14 x + 5 \right) d x$

A) $4 x ^ { 4 } - 7 x ^ { 2 } + 5 + C$
B) $4 x ^ { 4 } - 7 x + 5 x$
C) $12 x ^ { 3 } - 14 x ^ { 2 } + 5 x + C$
D) $4 x ^ { 4 } - 7 x ^ { 2 } + 5 x + C$
E) $4 x ^ { 4 } - 7 x ^ { 3 } + 5 x + C$

Question 5

A ball is thrown vertically upwards from a height of 10 ft with an initial velocity of 70 ft per second. How high will the ball go?

A)239.6875 ft
B)67.4219 ft
C)86.5625 ft
D)219.6875 ft
E)241.6875 ft

Question 6

Find the indefinite integral $\int v ^ { - 1 / 10 } d v$ and check your result by differentiation.

A) $\frac { 9 v ^ { \frac { 9 } { 10 } } } { 10 }$
B) $\frac { v ^ { \frac { 9 } { 10 } } } { 9 }$
C) $\frac { 11 v ^ { \frac { 11 } { 10 } } } { 9 }$
D) $\frac { 10 v ^ { \frac { 9 } { 10 } } } { 9 }$
E) $\frac { v ^ { \frac { 11 } { 10 } } } { 11 }$

Question 7

Identify u and $d u / d x$ for the integral $\int \left( 8 + \frac { 1 } { x ^ { 8 } } \right) ^ { 8 } \left( - \frac { 8 } { x ^ { 9 } } \right) d x$ .

A) $u = 8 + \frac { 1 } { x ^ { 8 } }$ and $d u / d x = \frac { 8 } { x ^ { 9 } }$
B) $u = \left( 8 + \frac { 1 } { x ^ { 8 } } \right) ^ { 7 }$ and $du / d x = - \frac { 8 } { x ^ { 9 } }$
C) $u = 8 + \frac { 1 } { x ^ { 8 } }$ and $d u/ d x = - \frac { 8 } { x ^ { 9 } }$
D) $u = 8 + \frac { 1 } { x ^ { 8 } }$ and $d u / d x = - \frac { 9 } { x ^ { 9 } }$
E) $u = \left( 8 + \frac { 1 } { x ^ { 8 } } \right) ^ { 7 }$ and $d u / d x = \frac { 8 } { x ^ { 9 } }$

Question 8

Find the indefinite integral $\int 4 d x$ and check your result by differentiation.

A) $4 x$
B) $4 x ^ { 2 }$
C)4
D) $8$
E) $8 x$

Question 9

Find the particular solution that satisfies the differential equation $f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - 5$ and initial condition $f ( 4 ) = - 16$ .

A) $f ( x ) = \frac { 1 } { 5 } x ^ { 2 } - 5 x$
B) $f ( x ) = \frac { 1 } { 7 } x ^ { 2 } + 5 x - 1260$
C) $f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - 5 x$
D) $f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - 5 x - 1260$
E) $f ( x ) = \frac { 1 } { 5 } x ^ { 2 } + 5 x$

Question 10

Find the indefinite integral and check the result by differentiation. $\int ( - 8 t + 3 ) d t$

A) $- 4 t ^ { 2 } + 3 t + C$
B) $- 8 t ^ { 2 } + 3 t + C$
C) $- 4 t ^ { 2 } + 3 t$
D) $- 8$
E)none of the above

Question 11

Identify $u$ and $d u / d x$ for the integral $\int \sqrt { 1 - x ^ { 11 } } \left( - 11 x ^ { 10 } \right) d x$ .

A) $u = 1 - x ^ { 11 }$ and $d u / d x = - 12 x$
B) $u = 1 - x ^ { 11 }$ and $d u / d x = - 11 x ^ { 10 }$
C) $u = \sqrt { 1 - x ^ { 11 } }$ and $d u / d x = - 11 x ^ { 10 }$
D) $u= 1 - x ^ { 11 }$ and $d u / d x = 11 x$
E) $u = \sqrt { 1 - x ^ { 11 } }$ and $d u / d x = 12 x$

Question 12

Evaluate the integral $\int \left( 3 + x ^ { 3 / 5 } \right) d x$ .

A) $3 x + \frac { 5 } { 8 } x ^ { 8 / 5 } + C$
B) $3 x + \frac { 8 } { 5 } x ^ { 8 / 5 } + C$
C) $\frac { 9 } { 2 } + \frac { 5 } { 8 } x ^ { 8 / 5 } + C$
D) $\frac { 3 } { 5 } x ^ { - 2 / 5 } + C$
E) $\frac { 3 } { 5 } x ^ { 8 /5 } + C$

Question 13

Evaluate the integral $\int \left( - 2 x ^ { 2 } - 9 x - 9 \right) d x$ .

A) $- 6 x ^ { 3 } - 18 x ^ { 2 } - 9 x + C$
B) $- \frac { 2 } { 3 } x ^ { 3 } - \frac { 9 } { 2 } x ^ { 2 } + \frac { 81 } { 2 } + C$
C) $- \frac { 2 } { 3 } x ^ { 3 } - \frac { 9 } { 2 } x ^ { 2 } + \frac { 81 } { 2 } x + C$
D) $- \frac { 2 } { 3 } x ^ { 3 } - \frac { 9 } { 2 } x ^ { 2 } - 9 x + C$
E) $- 4 x - 9 + C$

Question 14

An evergreen nursery sells a certain shrub after 4 years. The growth rate of the shrub is given by $d h / d t = 2.5 t + 6$ , where t is the time in years and h is the height in centimeters. The seedlings are 10 centimeters tall when planted (t = 0). How tall are the shrubs when they are sold?

A)36 centimeters
B)40 centimeters
C)44 centimeters
D)54 centimeters
E)70 centimeters

Question 15

Evaluate the integral $\int \left( x ^ { 6 } + 7 x ^ { 4 } - 3 \right) d x$ .

A) $\frac { 1 } { 7 } x ^ { 7 } + \frac { 7 } { 5 } x ^ { 5 } - \frac { 9 } { 2 } + C$
B) $\frac { 1 } { 7 } x ^ { 7 } + \frac { 7 } { 5 } x ^ { 5 } - \frac { 9 } { 2 } x + C$
C) $\frac { 1 } { 7 } x ^ { 7 } + \frac { 7 } { 5 } x ^ { 5 } - 3 x + C$
D) $\frac { 1 } { 6 } x ^ { 6 } + \frac { 7 } { 4 } x ^ { 4 } - 3 x + C$
E) $\frac { 1 } { 5 } x ^ { 5 } + \frac { 7 } { 3 } x ^ { 3 } - 3 x + C$

Question 16

Evaluate the integral $\int \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } \left( 40 x ^ { 4 } \right) d x$

A) $\frac { 1 } { 6 } \left( 8 x ^ { 5 } + 9 \right) ^ { 6 } + C$
B) $\frac { 1 } { 7 } \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } + C$
C) $\frac { 1 } { 7 } \left( 8 x ^ { 5 } + 9 \right) ^ { 8 } + C$
D) $\frac { 1 } { 8 } \left( 8 x ^ { 5 } + 9 \right) ^ { 8 } + C$
E) $\frac { 1 } { 8 } \left( 8 x ^ { 5 } + 9 \right) ^ { 7 } + C$

Question 17

Find a function that satisfies the conditions $f ^ { \prime \prime } ( x ) = x ^ { 5 } , f ^ { \prime } ( 0 ) = 9 , f ( 0 ) = 5$ .

A) $f ( x ) = \frac { 1 } { 6 } x ^ { 6 } + 9 x$
B) $f ( x ) = \frac { 1 } { 42 } x ^ { 7 } + 9 x + 5$
C) $f ( x ) = \frac { 1 } { 42 } x ^ { 6 } + 9 x + 5$
D) $f ( x ) = \frac { 1 } { 7 } x ^ { 7 } + 9 x$
E) $f ( x ) = \frac { 1 } { 42 } x ^ { 6 } + 5 x + 9$

Question 18

Evaluate the integral $\int \frac { 4 } { x ^ { 6 } } d x$ .

A) $4 \ln \left( x ^ { 6 } \right) + C$
B) $\frac { 4 } { 7 x ^ { 7 } } + C$
C) $- \frac { 4 } { 5 } \ln \left( x ^ { 5 } \right) + C$
D) $- \frac { 4 } { 7 x ^ { 7 } } + C$
E) $- \frac { 4 } { 5 x ^ { 5 } } + C$

Question 19

The graph of the derivative of a function is given below. Sketch the graphs of two functions that have the given derivative.

A)
B)
C)
D)
E)

Question 20

Find the indefinite integral of the following function and check the result by differentiation. $\int ( 1 + 4 z ) ^ { 3 } d z$

A) $4 ( 1 + 4 z ) ^ { 4 } + C$
B) $\frac { ( 1 + 4 z ) ^ { 4 } } { 3 } + C$
C) $\frac { ( 1 + 4 z ) ^ { 4 } } { 4 } + C$
D) $\frac { ( 1 + 4 z ) ^ { 4 } } { 16 } + C$
E)none of the above

Question 21

Find the equation of the function f whose graph passes through the point $\left( 0 , \frac { 7 } { 3 } \right)$ and whose derivative is $f ^ { \prime } ( x ) = x \sqrt { 1 - x ^ { 2 } }$ .

A) $f ( x ) = \frac { 1 } { 3 } \left[ 8 - \left( 1 - x ^ { 2 } \right) ^ { 1 / 2 } \right]$
B) $f ( x ) = \frac { 1 } { 3 } \left[ 8 - \left( 1 - x ^ { 2 } \right) ^ { 3 / 2 } \right]$
C) $f ( x ) = \frac { 1 } { 3 } \left[ 10 + \left( 1 - x ^ { 2 } \right) ^ { 2 / 3 } \right]$
D) $f ( x ) = \frac { 1 } { 3 } \left[ 10 + \left( 1 - x ^ { 2 } \right) ^ { 1 / 2 } \right]$
E) $f ( x ) = \frac { 1 } { 3 } \left[ 10 - \left( 1 - x ^ { 2 } \right) ^ { 2 / 3 } \right]$

Question 22

Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 5 x } { \left( x ^ { 2 } + 5 \right) ^ { 4 } } d x$

A) $\frac { - 5 } { 6 \left( x ^ { 2 } + 5 \right) ^ { 3 } } + C$
B) $\frac { - 5 } { 6 \left( x ^ { 2 } + 5 \right) ^ { 3 } } + C$ .
C) $\frac { - 2 } { 6 \left( x ^ { 2 } + 5 \right) ^ { 3 } } + C$
D) $\frac { - 5 } { 6 \left( x ^ { 2 } + 5 \right) ^ { 4 } } + C$
E) $\frac { - 5 } { 6 \left( x ^ { 2 } + 5 \right) ^ { - 3 } } + C$

Question 23

Find the indefinite integral. $\int \frac { x } { - 4 x ^ { 2 } + 4 } d x$

A) $\frac { 1 } { - 8 x } + C$
B) $\ln \left| - 4 x ^ { 2 } + 4 \right| + C$
C) $\frac { - 1 } { 8 } \ln \left| - 4 x ^ { 2 } + 4 \right| + C$
D) $\frac { \ln \left| - 4 x ^ { 2 } + 4 \right| } { - 4 x ^ { 2 } + 4 } + C$
E) $\ln \left| - 4 x ^ { 2 } - 4 \right| + C$

Question 24

Evaluate the integral $\int 210 e ^ { 0.1 x } d x$

A) $21 e ^ { 0.1 x } + C$
B) $190.9 e ^ { 1.1 x } + C$
C) $2100 e ^ { 0.1 x } + C$
D) $21 e ^ { - 0.9 x } + C$
E) $21 e ^ { 1.1 x } + C$

Question 25

Find the indefinite integral. $\int \frac { x ^ { 2 } } { 3 x ^ { 3 } + 10 } d x$

A) $\frac { 1 } { 9 } \ln \left| 3 x ^ { 3 } + 10 \right| + C$
B) $\ln \left| 3 x ^ { 3 } + 10 \right| + C$
C) $\frac { x ^ { 3 } } { 3 x ^ { 4 } + 10 x } + C$
D)integral does not exist
E)none of the above

Question 26

Find the indefinite integral. $\int \frac { ( \ln x ) ^ { 4 } } { x } d x$

A) $\frac { ( \ln x ) ^ { 4 } } { 4 } + C$
B) $4 ( \ln x ) ^ { 3 } + C$
C) $\frac { 5 \ln x } { x } + C$
D) $\frac { ( \ln x ) ^ { 5 } } { 5 } + C$
E)none of the above

Question 27

Evaluate the integral $\int x ^ { 7 } e ^ { 7 x ^ { 8 } } d x$

A) $\frac { 1 } { 8 } e ^ { 7 x ^ { 8 } } + C$
B) $56 e ^ { 7 x ^ { 8 } } + C$
C) $\frac { 1 } { 56 } e ^ { 7 x ^ { 8 } } + C$
D) $8 e ^ { 7 x ^ { 8 } } + C$
E) $\frac { 1 } { 56 } x ^ { 8 } e ^ { 7 x ^ { 8 } } + C$

Question 28

Use formal substitution to find the indefinite integral $\int \frac { x ^ { 4 } + 1 } { \sqrt { x ^ { 5 } + 5 x + 6 } } d x$ .

A) $\frac { 2 } { 5 } \left( x ^ { 5 } + 5 x + 6 \right) + C$
B) $\frac { 1 } { 5 } \sqrt { x ^ { 4 } + 5 x + 6 } + C$
C) $\frac { 2 } { 5 } \sqrt { x ^ { 4 } + 5 x + 6 } + C$
D) $\frac { 2 } { 5 } \sqrt { x ^ { 5 } + 5 x + 6 } + C$
E) $\frac { 1 } { 5 } \left( x ^ { 5 } + 5 x + 6 \right) + C$

Question 29

Find the indefinite integral of the following function and check the result by differentiation. $\int x ^ { 4 } \sqrt { \left( 4 + x ^ { 5 } \right) } d x$

A) $\frac { \left( 4 + x ^ { 5 } \right) ^ { \frac { 3 } { 2 } } } { 10 } + C$
B) $\frac { 2 \left( 4 + x ^ { 5 } \right) ^ { \frac { 2 } { 3 } } } { 15 } + C$
C) $\frac { \left( 4 + x ^ { 5 } \right) ^ { \frac { 3 } { 2 } } } { 15 } + C$
D) $\frac { 2 \left( 4 + x ^ { 5 } \right) ^ { \frac { 3 } { 2 } } } { 15 } + C$
E)none of the above

Question 30

Use the Log Rule to find the indefinite integral for $\int \frac { 1 } { 4 - 6 x } d x$ .

A) $- \frac { 1 } { 6 } \ln | 4 + 6 x | + C$
B) $\frac { 1 } { 4 } \ln | 4 - 6 x | + C$
C) $- \frac { 1 } { 6 } \ln | 4 - 6 x | + C$
D) $\frac { 1 } { 4 } \ln | 4 + 6 x | + C$
E) $\frac { 1 } { 6 } \ln | 4 - 6 x | + C$

Question 31

Evaluate the integral $\int ( 7 x + 2 ) ^ { 1 / 5 } d x$

A) $\frac { 5 } { 6 } ( 7 x + 2 ) ^ { 6 / 5 } + C$
B) $\frac { 6 } { 5 } ( 7 x + 2 ) ^ { - 4 / 5 } + C$
C) $\frac { 35 } { 6 } ( 7 x + 2 ) ^ { - 4 / 5 } + C$
D) $\frac { 42 } { 5 } ( 7 x + 2 ) ^ { 6 / 5 } + C$
E) $\frac { 5 } { 42 } ( 7 x + 2 ) ^ { 6 / 5 } + C$

Question 32

Find the indefinite integral of the following function and check the result by differentiation. $\int u ^ { 2 } \left( 5 + u ^ { 3 } \right) ^ { 3 } d u$

A) $12 \left( 5 + u ^ { 3 } \right) ^ { 4 } + C$
B) $\frac { \left( 5 + u ^ { 3 } \right) ^ { 4 } } { 12 } + C$
C) $\frac { \left( 5 + u ^ { 3 } \right) ^ { 4 } } { 4 } + C$
D) $\frac { \left( 5 + u ^ { 2 } \right) ^ { 4 } } { 12 } + C$
E)none of the above

Question 33

Find the indefinite integral. $\int \frac { x ^ { 2 } + 18 x + 2 } { x ^ { 3 } + 27 x ^ { 2 } + 6 x } d x$

A) $\frac { 1 } { 3 } \ln \left| x ^ { 3 } + 27 x ^ { 2 } + 6 x \right| + C$
B) $- \frac { 1 } { 3 } \ln \left| x ^ { 3 } + 27 x ^ { 2 } + 6 x \right| + C$
C) $\ln \left| x ^ { 3 } + 27 x ^ { 2 } + 6 x \right| + C$
D) $- 3 \ln \left| x ^ { 3 } + 27 x ^ { 2 } + 6 x \right| + C$
E) $- \ln \left| x ^ { 3 } + 27 x ^ { 2 } + 6 x \right| + C$

Question 34

Find the supply function $x = f ( p )$ that satisfies $\frac { d x } { d p } = p \sqrt { p ^ { 2 } - 25 }$ and the initial condition x = 700 when $p = \$ 13$ .

A) $x = \frac { 1 } { 3 } \left( p ^ { 2 } - 25 \right) ^ { 3 / 2 } + 124$
B) $x = \frac { 1 } { 3 } \left( p ^ { 2 } - 25 \right) ^ { 1 / 2 } + 696$
C) $x = \frac { 1 } { 3 } ( p - 5 ) + 124$
D) $x = \frac { 1 } { 5 } \left( p ^ { 2 } - 25 \right) ^ { 3 / 2 } + 127$
E) $x = \frac { 1 } { p - 1 } \left( p ^ { 2 } - 25 \right) ^ { 1 / 2 } + 699$

Question 35

Evaluate the integral $\int e ^ { 23 x } d x$

A) $\frac { 1 } { 23 } e ^ { 23 x } + C$
B) $23 e ^ {2 3 x } + C$
C) $\frac { 1 } { 24 } e ^ { 24 x } + C$
D) $23 e ^ { 22 x } + C$
E) $\frac { 1 } { 22 } e ^ { 2 2 x } + C$

Question 36

The marginal cost of a product is modeled by $\frac { d C } { d x } = \frac { 8 } { \sqrt { x + 1 } }$ , when x = 8, C = 40. Find the cost function.

A) $C ( x ) = 8 \sqrt { x + 1 } + - 8$
B) $C ( x ) = 16 ( x + 1 ) + - 6$
C) $C ( x ) = 16 \sqrt { x + 1 } + - 8$
D) $C ( x ) = 8 \sqrt { x + 1 } + - 8$ .
E) $C ( x ) = 8 ( x + 1 ) + - 6$

Question 37

Find the indefinite integral. $\int \frac { 7 x ^ { 2 } } { 9 - 8 x ^ { 3 } } d x$

A) $\frac { 24 } { 7 } \ln \left| 9 - 8 x ^ { 3 } \right| + C$
B) $\ln \left| 9 - 8 x ^ { 3 } \right| + C$
C) $7 \ln \left| 9 - 8 x ^ { 3 } \right| + C$
D) $- \frac { 7 } { 24 } \ln \left| 9 - 8 x ^ { 3 } \right| + C$
E) $- \frac { 1 } { 24 } \ln \left| 9 - 8 x ^ { 3 } \right| + C$

Question 38

Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 3 t ^ { 2 } } { \sqrt { t ^ { 3 } + 3 } } d t$

A) $2 \sqrt { t ^ { 3 } + 3 } + C$
B) $\sqrt { t ^ { 3 } + 3 } + C$
C) $2 \sqrt { t ^ { 2 } + 3 } + C$
D) $\frac { 1 } { 2 } \sqrt { t ^ { 3 } + 3 } + C$
E) $\frac { 1 } { 2 } \sqrt { t ^ { 2 } + 3 } + C$

Question 39

Evaluate the integral $\int \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } x ^ { 4 } d x$

A) $\frac { 1 } { 140 } \left( 4 x ^ { 5 } + 3 \right) ^ { 7 } + C$
B) $\frac { 20 } { 9 } \left( 4 x ^ { 5 } + 3 \right) ^ { 9 } + C$
C) $\frac { 5 } { 2 } \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } + C$
D) $\frac { 1 } { 180 } \left( 4 x ^ { 5 } + 3 \right) ^ { 9 } + C$
E) $\frac { 1 } { 160 } \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } + C$

Question 40

Find the indefinite integral. $\int e ^ { - 4 y } d y$

A) $- 4 e ^ { - 4 y } + C$
B) $- 4 e ^ { - 5 y } + C$
C) $- \frac { 1 } { 4 } e ^ { - 5 y } + C$
D) $- \frac { 1 } { 4 } e ^ { - 4 y } + C$
E) $- \frac { 1 } { 4 } e ^ { \frac { - 1 } { 4 } y } + C$

Question 41

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { - 3 } ^ { 3 } \sqrt { 9 - x ^ { 2 } } d x$

A) $9 \pi$
B) $\frac { 9 \pi } { 4 }$
C) $\frac { 9 } { 2 }$
D) $\frac { 9 \pi } { 2 }$
E)none of the above

Question 42

Find the area of the region bounded by the graphs of the algebraic functions. $\begin{array} { l } f ( x ) = x ^ { 2 } - 4 x \\g ( x ) = 0\end{array}$

A) $A = \frac { 17 } { 6 }$
B) $A = \frac { 16 } { 3 }$
C) $A = \frac { 11 } { 2 }$
D) $A = \frac { 32 } { 3 }$
E) $A = \frac { 3 } { 1 }$

Question 43

Use any basic integration formula or formulas to find the indefinite integral $\int e ^ { x } \sqrt { 2 - e ^ { x } } d x$ .

A) $\frac { 3 } { 2 } \left( 2 - e ^ { x } \right) ^ { 3 / 2 } + C$
B) $\frac { 3 } { 2 } \left( 4 - e ^ { x } \right) ^ { 2 / 3 } + C$
C) $- \frac { 3 } { 2 } \left( 4 - e ^ { x } \right) ^ { 3 / 2 } + C$
D) $\frac { 2 } { 3 } \left( 2 - e ^ { x } \right) ^ { 2 / 3 } + C$
E) $- \frac { 2 } { 3 } \left( 2 - e ^ { x } \right) ^ { 3 / 2 } + C$

Question 44

Find the area between the curve $y = 5 x ^ { 2 } - 6 x - 8$ and the x-axis from $x = - 1 \text { to } x = 3$ .

A) $\frac { 28 } { 3 }$
B) $\frac { 196 } { 3 }$
C) $\frac { 140 } { 9 }$
D) $\frac { 140 } { 3 }$
E) $\frac { 14 } { 3 }$

Question 45

Set up the definite integral that gives the area of the region bounded by the graphs. $\begin{array} { l } f ( x ) = ( x - 4 ) ^ { 3 } \\g ( x ) = x - 4\end{array}$

A) $\int _ { 3 } ^ { 4 } \left( ( x - 4 ) - ( x - 4 ) ^ { 3 } \right) d x + \int _ { 4 } ^ { 5 } \left( ( x - 4 ) ^ { 3 } - ( x - 4 ) \right) d x$
B) $\int _ { - 1 } ^ { 0 } \left( ( x - 4 ) + ( x - 4 ) ^ { 3 } \right) d x + \int _ { 0 } ^ { 1 } \left( ( x - 4 ) ^ { 3 } + ( x - 4 ) \right) d x$
C) $\int _ { - 1 } ^ { 0 } \left( ( x - 4 ) ^ { 3 } - ( x - 4 ) \right) d x + \int _ { 0 } ^ { 1 } \left( ( x - 4 ) - ( x - 4 ) ^ { 3 } \right) d x$
D) $\int _ { 3 } ^ { 4 } \left( ( x - 4 ) ^ { 3 } - ( x - 4 ) \right) d x + \int _ { 4 } ^ { 5 } \left( ( x - 4 ) - ( x - 4 ) ^ { 3 } \right) d x$
E) $\int _ { 3 } ^ { 4 } \left( ( x - 4 ) ^ { 3 } + ( x - 4 ) \right) d x + \int _ { 4 } ^ { 5 } \left( ( x - 4 ) + ( x - 4 ) ^ { 3 } \right) d x$

Question 46

Find the average value of the function over the given interval. $f ( x ) = 7 \sqrt [ 3 ] { x }$ on [0,8]

A) $\frac { 7 } { 4 }$
B) $\frac { 14 } { 9 }$
C) $\frac { 21 } { 2 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 64 } { 3 }$

Question 47

Evaluate the following definite integral. $\int _ { 1 } ^ { 4 } \frac { 1 } { \sqrt { 6 z + 7 } } d z$ Use a graphing utility to check your answer.

A) $\frac { \sqrt { 31 } - \sqrt { 13 } } { 6 }$
B) $\frac { \sqrt { 31 } - \sqrt { 13 } } { 3 }$
C) $\frac { \sqrt { 31 } + \sqrt { 13 } } { 3 }$
D) $\frac { \sqrt { 31 } + \sqrt { 13 } } { 6 }$
E) $\frac { \sqrt { 13 } - \sqrt { 31 } } { 3 }$

Question 48

Determine the area of the given region. $$y = 2 x ( 1 - x )$$

A) $\frac { 5 } { 3 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 3 } { 7 }$
D) $\frac { 1 } { 2 }$
E)None of the above

Question 49

Find the area of the region bounded by the graphs of the algebraic functions. $\begin{array} { l } f ( x ) = x ^ { 2 } + 18 x + 81 \\g ( x ) = 11 ( x + 9 )\end{array}$

A) $A = \frac { 1331 } { 6 }$
B) $A = \frac { 1331 } { 3 }$
C) $A = \frac { 1331 } { 12 }$
D) $A = \frac { 1493 } { 6 }$
E) $A = \frac { 1817 } { 6 }$

Question 50

The rate of depreciation of a building is given by $D ^ { \prime } ( t ) = 8,300 ( 20 - t )$ dollars per year, $0 \leq t \leq 20$ Use the definite integral to find the total depreciation over the first $20$ years.

A) $\$ 1,660,000$
B) $\$ 83,000$
C) $\$ 830,000$
D) $\$ 474,286$
E) $\$ 3,320,000$

Question 51

Determine the graph whose area (the shaded region) is represented by the integral. $\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 4 x + 5 \right) - ( x + 1 ) d x$

A)
B)
C)
D)
E)

Question 52

Find the average value of the function over the given interval. $f ( x ) = 9 - x ^ { 2 }$ on [-3,3]

A)6
B)21
C)36
D)4
E)20

Question 53

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 3 } ^ { 7 } 5 t ~d t$

A)-145
B)290
C)200
D)100
E)10

Question 54

The integrand of the following definite integral is a difference of two functions. $$\int _ { 0 } ^ { 4 } \left[ ( x + 1 ) - \frac { 1 } { 2 } x \right] d x$$ Sketch the graph of the two functions and shade the region whose area is represented by the integral.

A)
B)
C)
D)
E)

Question 55

Evaluate the definite integral of the algebraic function. $\int _ { 2 } ^ { 4 } ( - 6 z + 4 ) d z$ Use a graphing utility to verify your results.

A)-12
B)-56
C)-36
D)8
E)-28

Question 56

Find the area of the shaded region.

A) $\frac { 13 } { 6 }$
B) $\frac { 37 } { 12 }$
C) $\frac { 37 } { 6 }$
D) $\frac { 13 } { 12 }$
E) $\frac { 13 } { 7 }$

Question 57

Use the values $\int _ { 0 } ^ { 2 } f ( x ) d x = 8$ and $\int _ { 0 } ^ { 2 } g ( x ) d x = 3$ to evaluate the definite integral $\int _ { 0 } ^ { 2 } [ f ( x ) - 2 g ( x ) ] d x$ .

A)14
B)2
C)5
D)11
E)4

Question 58

Find the equation of the function whose derivative is $f ^ { \prime } ( x ) = \frac { x ^ { 2 } + 9 x + 7 } { x - 1 }$ and whose graph passes through the point $( 2,4 )$ .

A) $\frac { x ^ { 2 } } { 2 } + 10 x + 17 \ln | x - 1 | - 19$
B) $\frac { x ^ { 2 } } { 2 } + 9 x + 17 \ln | x - 1 | - 18$
C) $\frac { x ^ { 2 } } { 2 } + 10 x + 17 \ln | x - 1 | - 18$
D) $\frac { x ^ { 2 } } { 2 } + 9 x + 7 \ln | x - 1 | - 18$
E) $\frac { x ^ { 2 } } { 2 } + 10 x + 7 \ln | x - 1 | - 19$

Question 59

Evaluate the definite integral $\int _ { 4 } ^ { 7 } ( x - 7 ) ^ { 5 } d x$ .

A) $\frac { 729 } { 5 }$
B) $- \frac { 729 } { 7 }$
C) $- \frac { 729 } { 5 }$
D) $\frac { 243 } { 2 }$
E) $- \frac { 243 } { 2 }$

Question 60

Evaluate the definite integral of the algebraic function. $\int _ { 3 } ^ { 8 } \left( u ^ { \frac { 6 } { 5 } } - u ^ { \frac { 5 } { 6 } } \right) d u$ Use a graphing utility to verify your results.

A)18.4007
B)29.6329
C)59.5941
D)38.9974
E)48.0336

Question 61

Estimate the surface area of the pond shown in the figure using the Midpoint Rule.

A) $\approx 990 \mathrm { ft } ^ { 2 }$
B) $\approx 9920 \mathrm {f t } ^ { 2 }$
C) $\approx 9020 \mathrm {f t } ^ { 2 }$
D) $\approx 9990 \mathrm { ft } ^ { 2 }$
E) $\approx 920 \mathrm { ft } ^ { 2 }$

Question 62

Use the Midpoint Rule with n = 4 to approximate the area of the following region. $f ( x ) = \frac { 1 } { x } , [ 1,5 ]$

A)1.156
B)1.324
C)1.575
D)1.275
E)1.876

Question 63

The demand function for a product is $p = 140 - 4 x$ , where p is the number of dollars and x is the number of units. If the equilibrium price is $\$ 40$ , what is the consumer's surplus?

A)$ $1175$
B)$ $1250$
C)$ $1375$
D)$ $1220$
E)$ $1340$

Question 64

Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of $f ( x ) = 4 - x ^ { 2 }$ and the x-axis over the interval [ $- 2,2$ ].

A)11.4421
B)11.0023
C)11.2114
D)11.0000
E)12.0000

Question 65

The revenue from a manufacturing process (in millions of dollars per year) is projected to follow the model $R = 400 + 0.07 t$ for 10 years. Over the same period of time, the cost (in millions of dollars per year) is projected to follow the model $C = 60 + 0.5 t ^ { 2 }$ , where t is the time (in years). Approximate the profit over the 10-year period, beginning with t = 0. Round your answer to two decimal places.

A) $\$ 3236.83$ million
B) $\$ 3378.50$ million
C) $\$ 3153.50$ million
D) $\$ 3235.67$ million
E) $\$ 3385.67$ million

Question 66

Use the Midpoint Rule n = 4 to approximate the area of the following region. $f ( y ) = \frac { 1 } { 4 } y , [ 2,4 ]$

A)2.5
B)1.2
C)1.5
D)1.9
E)2.3

Question 67

Estimate the surface area of the oil spill shown in the figure using the Midpoint Rule.

A) $\approx$ 481.6 $m i ^ { 2 }$
B) $\approx$ 301.6 $m i ^ { 2 }$
C) $\approx$ 311.6 $m i ^ { 2 }$
D) $\approx$ 431.6 $m i ^ { 2 }$
E) $\approx$ 381.6 $m i ^ { 2 }$

Question 68

Use the Midpoint Rule with $n = 4$ to approximate the area of the region bounded by the graph of and the -axis over the interval. Sketch the region. $f ( x ) = \left( x ^ { 2 } - 4 \right) ^ { 2 } , [ - 2,2 ]$

A)The approximate area is: $\approx 30.25$
B)The approximate area is: $\approx 24.25$
C)The approximate area is: $\approx 34.25$
D)The approximate area is: $\approx 14.25$
E)The approximate area is: $\approx 34.99$

Question 69

Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of $f ( x ) = 3 x - x ^ { 4 }$ and the x-axis over the interval [0,1].

A)1.7524
B)1.3126
C)1.5217
D)2.3103
E)1.3103

Question 70

Two models, $R _ { 1 } = 7.21 + 0.58 t$ and $R _ { 2 } = 7.21 + 0.43 t$ , are given for revenue (in billions of dollars per year) for a large corporation. Both models are estimates of revenues for 2007 through 2011, with t = 7 corresponding to 2007. Which model is projecting the greater revenue? How much more total revenue does that model project over the five-year period?

A)The model $R _ { 1 }$ projects greater revenue than $R _ { 2 }$ . $\$ 8.75$ billion
B)The model $R _ { 2 }$ projects greater revenue than $R _ { 1 }$ . $\$ 7.75$ billion
C)The model $R _ { 1 }$ projects greater revenue than $R _ { 2 }$ . $\$ 6.75$ billion
D)The model $R _ { 1 }$ projects greater revenue than $R _ { 2 }$ . $\$ 10.75$ billion
E)The model $R _ { 2 }$ projects greater revenue than $R _ { 1 }$ . $\$ 16.75$ billion

Question 71

Find the consumer and producer surpluses by using the demand and supply functions, where p is the price (in dollars) and x is the number of units (in millions). Demand Function $~~~~~~~~~~~~~~~~$ Supply Function
$P = 975 - 23 x ~~~~~~~~~~~\quad 42 x$

A)a. $2587.50b. $3725.00
B)a. $5587.50b. $4725.00
C)a. $2587.50b. $1725.00
D)a. $1587.50b. $4725.00
E)a. $3587.50b. $4725.00

Question 72

Estimate the surface area of the golf green shown in the figure using the midpoint rule.

A)966
B)161
C)1449
D)1550
E)234

Question 73

Use the Midpoint Rule with $n = 4$ to approximate $\pi$ where $\pi = \int _ { 0 } ^ { 1 } \frac { 4 } { 1 + x ^ { 2 } } d x$ . Then use a graphing utility to evaluate the definite integral. Compare your results.

A)a. Midpoint Rule: $\approx 0.146801$ b. Graphing utility: $\approx 3.141593$
B)a. Midpoint Rule: $\approx 3.146801$ b. Graphing utility: $\approx 0.141593$
C)a. Midpoint Rule: $\approx 1.146801$ b. Graphing utility: $\approx 3.141593$
D)a. Midpoint Rule: $\approx 3.146801$ b. Graphing utility: $\approx 3.141593$
E)a. Midpoint Rule: $\approx 3.146801$ b. Graphing utility: $\approx 1.141593$

Question 74

Use the rectangles to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$f ( x ) = - 2 x + 3 , [ 0,1 ]$$

A)a. The approximate area: 3b. The exact area: 2
B)a. The approximate area: 2b. The exact area: 3
C)a. The approximate area: 2b. The exact area: 1
D)a. The approximate area: 2b. The exact area: 2
E)a. The approximate area: 1b. The exact area: 2