Question 1

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

Accepted Answer

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)  $\frac { 1 } { 2 } x ^ { 2 } + 28 \sqrt { x } + C$ 
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)  $\frac { 1 } { 2 } x ^ { 2 } + 28 \sqrt { x } + C$ B

Question 2

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

Accepted Answer

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 $d u / d x = 12 x$ 
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 $d u / d x = 12 x$ B

Question 3

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

Accepted Answer

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)  $\approx 920 \mathrm { ft } ^ { 2 }$ 
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)  $\approx 920 \mathrm { ft } ^ { 2 }$ B

Question 4

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

Accepted Answer

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)  $\frac { 1 } { 6 } \ln | 4 - 6 x | + C$ 
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 5

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

Accepted Answer

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

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

Accepted Answer

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 $d u / d x = \frac { 8 } { x ^ { 9 } }$ 
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 7

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

Accepted Answer

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 
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 8

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.

Accepted Answer

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: $\approx 1.141593$ 
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 9

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 $( 2,4 )$ .

Accepted Answer

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)  $\frac { x ^ { 2 } } { 2 } + 10 x + 7 \ln | x - 1 | - 19$ 
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 10

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

Accepted 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)  $\frac { \sqrt { 13 } - \sqrt { 31 } } { 3 }$ 
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 11

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

Accepted Answer

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)  $- \ln \left| x ^ { 3 } + 27 x ^ { 2 } + 6 x \right| + C$ 
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 12

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

Accepted Answer

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 
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 13

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 [ $- 2,2$ ].

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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)  $\frac { 3 } { 5 } x ^ { 8 /5 } + C$ 
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 16

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.

Accepted Answer

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

Question 17

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

Accepted Answer

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)  $\int _ { 3 } ^ { 4 } \left( ( x - 4 ) ^ { 3 } + ( x - 4 ) \right) d x + \int _ { 4 } ^ { 5 } \left( ( x - 4 ) + ( x - 4 ) ^ { 3 } \right) d x$ 
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 18

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).

Accepted Answer

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)  $C ( x ) = \frac { 1 } { 120 } x ^ { 4 } + 90 x + 2000$ 
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)  $C ( x ) = \frac { 1 } { 120 } x ^ { 4 } + 90 x + 2000$

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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)  $\frac { 19 } { 7 } x ^ { 7 } + C$ 
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)  $\frac { 19 } { 7 } x ^ { 7 } + C$

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

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

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

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

Find the area of the region bounded by the graphs of the algebraic functions. f(x)=+18x+81 g(x)=11(x+9)

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

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

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.

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 $( 2,4 )$ .

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

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

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

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 [ $- 2,2$ ].

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

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

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

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).

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

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

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Exam 12: Integration and Its Applications

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

Identify uuu and du/dxd u / d xdu/dx for the integral ∫1−x11(−11x10)dx\int \sqrt { 1 - x ^ { 11 } } \left( - 11 x ^ { 10 } \right) d x∫1−x11​(−11x10)dx .

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

Use the Log Rule to find the indefinite integral for ∫14−6xdx\int \frac { 1 } { 4 - 6 x } d x∫4−6x1​dx .

Find the area of the region bounded by the graphs of the algebraic functions. f(x)=+18x+81 g(x)=11(x+9)

Identify u and du/dxd u / d xdu/dx for the integral ∫(8+1x8)8(−8x9)dx\int \left( 8 + \frac { 1 } { x ^ { 8 } } \right) ^ { 8 } \left( - \frac { 8 } { x ^ { 9 } } \right) d x∫(8+x81​)8(−x98​)dx .

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

Use the Midpoint Rule with n=4n = 4n=4 to approximate π\piπ where π=∫0141+x2dx\pi = \int _ { 0 } ^ { 1 } \frac { 4 } { 1 + x ^ { 2 } } d xπ=∫01​1+x24​dx . Then use a graphing utility to evaluate the definite integral. Compare your results.

Find the equation of the function whose derivative is f′(x)=x2+9x+7x−1f ^ { \prime } ( x ) = \frac { x ^ { 2 } + 9 x + 7 } { x - 1 }f′(x)=x−1x2+9x+7​ and whose graph passes through the point (2,4)( 2,4 )(2,4) .

Evaluate the following definite integral. ∫1416z+7dz\int _ { 1 } ^ { 4 } \frac { 1 } { \sqrt { 6 z + 7 } } d z∫14​6z+7​1​dz Use a graphing utility to check your answer.

Find the indefinite integral. ∫x2+18x+2x3+27x2+6xdx\int \frac { x ^ { 2 } + 18 x + 2 } { x ^ { 3 } + 27 x ^ { 2 } + 6 x } d x∫x3+27x2+6xx2+18x+2​dx

Find the indefinite integral of the following function and check the result by differentiation. ∫x4(4+x5)dx\int x ^ { 4 } \sqrt { \left( 4 + x ^ { 5 } \right) } d x∫x4(4+x5)​dx

Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f(x)=4−x2f ( x ) = 4 - x ^ { 2 }f(x)=4−x2 and the x-axis over the interval [ −2,2- 2,2−2,2 ].

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

Evaluate the integral ∫(3+x3/5)dx\int \left( 3 + x ^ { 3 / 5 } \right) d x∫(3+x3/5)dx .

Set up the definite integral that gives the area of the region bounded by the graphs. f(x)=(x-4 g(x)=x-4

Find the cost function for the marginal cost dCdx=140x2+90\frac { d C } { d x } = \frac { 1 } { 40 } x ^ { 2 } + 90dxdC​=401​x2+90 and fixed cost of $2000\$ 2000$2000 (for x = 0).

Find the area between the curve y=5x2−6x−8y = 5 x ^ { 2 } - 6 x - 8y=5x2−6x−8 and the x-axis from x=−1 to x=3x = - 1 \text { to } x = 3x=−1 to x=3 .

Evaluate the integral ∫19x7dx\int 19 x ^ { 7 } d x∫19x7dx .

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

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

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

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

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

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.

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 $( 2,4 )$ .

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

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

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

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 [ $- 2,2$ ].

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

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

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

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).

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

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