Question 1

Evaluate.
-$\int\left(t^{3}+e^{2 t}\right) d t$

Accepted Answer

A)  $\frac{t^{4}}{4}+\frac{e^{2 t}}{2}+C$ 
B)  $\frac{t^{2}}{2}+2 e^{2 t}+C$ 
C)  $\frac{\mathrm{t}^{4}}{4}+\frac{\mathrm{e}^{3 \mathrm{t}}}{3}+\mathrm{C}$ 
D)  $\frac{t^{4}}{4}+e^{2 t}+C$ 
A)  $\frac{t^{4}}{4}+\frac{e^{2 t}}{2}+C$ 
B)  $\frac{t^{2}}{2}+2 e^{2 t}+C$ 
C)  $\frac{\mathrm{t}^{4}}{4}+\frac{\mathrm{e}^{3 \mathrm{t}}}{3}+\mathrm{C}$ 
D)  $\frac{t^{4}}{4}+e^{2 t}+C$

Question 2

Find the area bounded by the given curves.
-$y=x^{2}, y=4$

Accepted Answer

A)  $\frac{37}{3}$ 
B)  $\frac{31}{3}$ 
C)  $\frac{32}{3}$ 
D)  $\frac{34}{3}$ 
A)  $\frac{37}{3}$ 
B)  $\frac{31}{3}$ 
C)  $\frac{32}{3}$ 
D)  $\frac{34}{3}$

Question 3

Find the integral.
-$\int(5 x-3)^{2} d x$

Accepted Answer

A)  $\frac{25}{3} x^{3}+9 x+C$ 
B)  $\frac{5}{2} x^{2}+9 x+C$ 
C)  $\frac{5}{2} x^{3}-15 x^{2}+9 x+C$ 
D)  $\frac{25}{3} x^{3}-15 x^{2}+9 x+C$ 
A)  $\frac{25}{3} x^{3}+9 x+C$ 
B)  $\frac{5}{2} x^{2}+9 x+C$ 
C)  $\frac{5}{2} x^{3}-15 x^{2}+9 x+C$ 
D)  $\frac{25}{3} x^{3}-15 x^{2}+9 x+C$

Question 4

Find the integral.
-$\int \frac{1}{8-x} d x$

Accepted Answer

A)  $-\ln |8-x|+C$ 
B)  $\ln (8-x)+C$ 
C)  $\ln |8-x|+C$ 
D)  $-\frac{1}{8} \ln |8-x|+C$ 
A)  $-\ln |8-x|+C$ 
B)  $\ln (8-x)+C$ 
C)  $\ln |8-x|+C$ 
D)  $-\frac{1}{8} \ln |8-x|+C$

Question 5

Find the integral.
-$\int \frac{3 \mathrm{e}^{\sqrt{\mathrm{z}}}}{8 \sqrt{\mathrm{z}}} \mathrm{dz}$

Accepted Answer

A)  $-12 e^{\sqrt{z}}+C$ 
B)  $-24 e^{\sqrt{z}}+C$ 
C)  $\frac{3}{4} \mathrm{e}^{\sqrt{\mathrm{z}}}+\mathrm{C}$ 
D)  $\frac{3}{8} e^{\sqrt{z}}+C$ 
A)  $-12 e^{\sqrt{z}}+C$ 
B)  $-24 e^{\sqrt{z}}+C$ 
C)  $\frac{3}{4} \mathrm{e}^{\sqrt{\mathrm{z}}}+\mathrm{C}$ 
D)  $\frac{3}{8} e^{\sqrt{z}}+C$

Question 6

Find the area of the shaded region.
-

Accepted Answer

A)  $\frac{17}{4}$ 
B)  $\frac{9}{4}$ 
C)  $\frac{41}{4}$ 
D)  $\frac{33}{4}$ 
A)  $\frac{17}{4}$ 
B)  $\frac{9}{4}$ 
C)  $\frac{41}{4}$ 
D)  $\frac{33}{4}$

Question 7

Solve the problem.
-Suppose that the acceleration of an object is given by $a(t)=5 t^{2 / 3}+2 e^{-t}$. The velocity of the object at time 0 is $v(0)=5$ and its position at time 0 is $s(0)=-9$. Find $s(t)$.

Accepted Answer

A)  $\frac{9 t^{8 / 3}}{8}+2 e^{-t}+5 t-9$ 
B)  $\frac{9 t^{8 / 3}}{8}+2 e^{-t}+7 t-11$ 
C)  $\frac{200 t^{8 / 3}}{9}+2 e^{-t}+3 t-11$ 
D)  $3 t^{5 / 3}-2 e^{-t}+7$ 
A)  $\frac{9 t^{8 / 3}}{8}+2 e^{-t}+5 t-9$ 
B)  $\frac{9 t^{8 / 3}}{8}+2 e^{-t}+7 t-11$ 
C)  $\frac{200 t^{8 / 3}}{9}+2 e^{-t}+3 t-11$ 
D)  $3 t^{5 / 3}-2 e^{-t}+7$

Question 8

Evaluate the integral.
-$\int_{-3}^{-1}\left(x^{2}+x+8\right) d x$

Accepted Answer

A)  20.67 
B)  20.5 
C)  36.33 
D)  21.17 
A)  20.67 
B)  20.5 
C)  36.33 
D)  21.17

Question 9

Evaluate the definite integral.
-$\int_{-3}^{1} \frac{x}{x^{2}+7} d x$

Accepted Answer

A)  $\frac{1}{2} \ln 2$ 
B)  $-\frac{1}{2} \ln 16$ 
C)  $\frac{1}{2} \ln \frac{1}{2}$ 
D)  $\frac{1}{2} \ln 8$ 
A)  $\frac{1}{2} \ln 2$ 
B)  $-\frac{1}{2} \ln 16$ 
C)  $\frac{1}{2} \ln \frac{1}{2}$ 
D)  $\frac{1}{2} \ln 8$

Question 10

Provide an appropriate response.
-If $f(x) \geq 0$ on the interval $[a, b]$, then $\int_{a}^{b} f(x) d x$ represents
i. the area to the right of the $y$-axis between $y=a$ and $y=b$.
ii. the area above the $x$-axis between $x=a$ and $x=b$.
iii. the area below the $x$-axis between $x=a$ and $x=b$.

Accepted Answer

A)  None of the above is correct. 
B)  Both ii and iii could be correct. 
C)  Only ii is correct. 
D)  Only $i$ is correct. 
A)  None of the above is correct. 
B)  Both ii and iii could be correct. 
C)  Only ii is correct. 
D)  Only $i$ is correct.

Question 11

Approximate the area under the curve and above the $x$-axis using $n$ rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle.
-$f(x)=2 x^{2}+x+3$ from $x=0$ to $x=6 ; n=6$

Accepted Answer

A)  221 
B)  211 
C)  230 
D)  200 
A)  221 
B)  211 
C)  230 
D)  200

Question 12

The given marginal function has x measured in years and MR(x) in hundreds of dollars per year. Use numerical integration on a graphing calculator to find the total revenue over the given period.
-$\operatorname{MR}(\mathrm{x})=0.03 \mathrm{x}^{3}-0.4 \mathrm{x}^{2}+2 \mathrm{x} \quad(0 \leq \mathrm{x} \leq 10)$

Accepted Answer

A)  $\$ 41.67$ 
B)  $\$ 107.72$ 
C)  $\$ 4166.67$ 
D)  $-\$ 5833.33$ 
A)  $\$ 41.67$ 
B)  $\$ 107.72$ 
C)  $\$ 4166.67$ 
D)  $-\$ 5833.33$

Question 13

Solve the problem.
-The rate of expenditure of a particular machine is given by $M^{\prime}(x)=15 x \sqrt{x^{2}+5}$, where $x$ is time measured in years. Maintenance costs through the second year are $\$ 125$. Find the total maintenance function.

Accepted Answer

A) M $(x)=5\left(x^{2}+5\right)^{3 / 2}+110$ 
B)  $M(x)=15\left(x^{2}+5\right)^{3 / 2}-10$ 
C) M $(\mathrm{x})=15\left(\mathrm{x}^{2}+5\right)^{3 / 2}+110$ 
D)  $M(x)=5\left(x^{2}+5\right)^{3 / 2}-10$ 
A) M $(x)=5\left(x^{2}+5\right)^{3 / 2}+110$ 
B)  $M(x)=15\left(x^{2}+5\right)^{3 / 2}-10$ 
C) M $(\mathrm{x})=15\left(\mathrm{x}^{2}+5\right)^{3 / 2}+110$ 
D)  $M(x)=5\left(x^{2}+5\right)^{3 / 2}-10$

Question 14

Evaluate.
-$\int x^{3} d x$

Accepted Answer

A)  $3 x^{2}+C$ 
B)  $4 x^{4}+C$ 
C)  $\frac{1}{4} x^{4}+C$ 
D)  $\frac{1}{3} x^{2}+C$ 
A)  $3 x^{2}+C$ 
B)  $4 x^{4}+C$ 
C)  $\frac{1}{4} x^{4}+C$ 
D)  $\frac{1}{3} x^{2}+C$

Question 15

Provide the proper response.
-Areas of complicated regions are estimated by filling in with shapes whose area is easy to compute. The shape that is most often used in defining the definite integral is ?.

Accepted Answer

A)  the circle 
B)  the triangle 
C)  the square 
D)  the rectangle 
A)  the circle 
B)  the triangle 
C)  the square 
D)  the rectangle

Question 16

Use integration by parts to find the integral.
-$\int x^{2} \ln 7 x d x$

Accepted Answer

A)  $\frac{1}{3} x^{3} \ln 7 x-\frac{1}{9} x^{3}+C$ 
B)  $\ln 7 x-\frac{1}{3} x^{3}+C$ 
C)  $\frac{1}{3} x^{3} \ln 7 x+\frac{1}{9} x^{3}+C$ 
D)  $\frac{1}{3} x^{3} \ln 7 x-\frac{1}{12} x^{4}+C$ 
A)  $\frac{1}{3} x^{3} \ln 7 x-\frac{1}{9} x^{3}+C$ 
B)  $\ln 7 x-\frac{1}{3} x^{3}+C$ 
C)  $\frac{1}{3} x^{3} \ln 7 x+\frac{1}{9} x^{3}+C$ 
D)  $\frac{1}{3} x^{3} \ln 7 x-\frac{1}{12} x^{4}+C$

Question 17

Use integration by parts to find the integral.
-$\int(6 x+3) e^{-2 x} d x$

Accepted Answer

A)  $-3 x e^{-2 x}-e^{-2 x}+C$ 
B)  $-3 x e^{-2 x}-3 e^{-2 x}+C$ 
C)  $3 x e^{-2 x}+3 e^{-2 x}+C$ 
D)  $-12 x e^{-2 x}-30 e^{-2 x}+C$ 
A)  $-3 x e^{-2 x}-e^{-2 x}+C$ 
B)  $-3 x e^{-2 x}-3 e^{-2 x}+C$ 
C)  $3 x e^{-2 x}+3 e^{-2 x}+C$ 
D)  $-12 x e^{-2 x}-30 e^{-2 x}+C$

Question 18

Evaluate.
-$\int \sqrt{\mathrm{x}}+\frac{\mathrm{x}}{\mathrm{x}^{5}} \mathrm{dx}$

Accepted Answer

A)  $2 x \sqrt{x}-\frac{1}{3} x^{-3}+C$ 
B)  $\frac{2}{3} x^{3 / 2}-\frac{1}{4} x^{-4}+C$ 
C)  $\frac{2}{3} x^{3 / 2}-\frac{1}{3} x^{-3}+C$ 
D)  $2 x \sqrt{x}-\frac{1}{4} x^{-4}+C$ 
A)  $2 x \sqrt{x}-\frac{1}{3} x^{-3}+C$ 
B)  $\frac{2}{3} x^{3 / 2}-\frac{1}{4} x^{-4}+C$ 
C)  $\frac{2}{3} x^{3 / 2}-\frac{1}{3} x^{-3}+C$ 
D)  $2 x \sqrt{x}-\frac{1}{4} x^{-4}+C$

Question 19

Use the definite integral to find the area between the $x$-axis and the graph of $f(x)$ over the indicated interval.
-$\mathrm{f}(\mathrm{x})=\frac{1}{\sqrt{\mathrm{x}}} ;[1,4]$

Accepted Answer

A)  4 
B)  $\frac{1}{2}$ 
C)  2 
D)  $\frac{1}{4}$ 
A)  4 
B)  $\frac{1}{2}$ 
C)  2 
D)  $\frac{1}{4}$

Question 20

Evaluate.
-$\int 9 \sqrt[3]{x} d x$

Accepted Answer

A)  $27 x^{4 / 3}+C$ 
B)  $\frac{27}{4} x^{4 / 3}+C$ 
C)  $\frac{9}{4} x^{4 / 3}+C$ 
D)  $9 x^{4 / 3}+C$ 
A)  $27 x^{4 / 3}+C$ 
B)  $\frac{27}{4} x^{4 / 3}+C$ 
C)  $\frac{9}{4} x^{4 / 3}+C$ 
D)  $9 x^{4 / 3}+C$

Evaluate. - $\int\left(t^{3}+e^{2 t}\right) d t$

Find the area bounded by the given curves. - $y=x^{2}, y=4$

Find the integral. - $\int(5 x-3)^{2} d x$

Find the integral. - $\int \frac{1}{8-x} d x$

Find the integral. - $\int \frac{3 \mathrm{e}^{\sqrt{\mathrm{z}}}}{8 \sqrt{\mathrm{z}}} \mathrm{dz}$

Find the area of the shaded region. -

Solve the problem. -Suppose that the acceleration of an object is given by $a(t)=5 t^{2 / 3}+2 e^{-t}$ . The velocity of the object at time 0 is $v(0)=5$ and its position at time 0 is $s(0)=-9$ . Find $s(t)$ .

Evaluate the integral. - $\int_{-3}^{-1}\left(x^{2}+x+8\right) d x$

Evaluate the definite integral. - $\int_{-3}^{1} \frac{x}{x^{2}+7} d x$

Approximate the area under the curve and above the $x$ -axis using $n$ rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - $f(x)=2 x^{2}+x+3$ from $x=0$ to $x=6 ; n=6$

Solve the problem. -The rate of expenditure of a particular machine is given by $M^{\prime}(x)=15 x \sqrt{x^{2}+5}$ , where $x$ is time measured in years. Maintenance costs through the second year are $\$ 125$ . Find the total maintenance function.

Evaluate. - $\int x^{3} d x$

Provide the proper response. -Areas of complicated regions are estimated by filling in with shapes whose area is easy to compute. The shape that is most often used in defining the definite integral is ?.

Use integration by parts to find the integral. - $\int x^{2} \ln 7 x d x$

Use integration by parts to find the integral. - $\int(6 x+3) e^{-2 x} d x$

Evaluate. - $\int \sqrt{\mathrm{x}}+\frac{\mathrm{x}}{\mathrm{x}^{5}} \mathrm{dx}$

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $\mathrm{f}(\mathrm{x})=\frac{1}{\sqrt{\mathrm{x}}} ;[1,4]$

Evaluate. - $\int 9 \sqrt[3]{x} d x$

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Exam 13: Integral Calculus

Evaluate. - ∫(t3+e2t)dt\int\left(t^{3}+e^{2 t}\right) d t∫(t3+e2t)dt

Find the area bounded by the given curves. - y=x2,y=4y=x^{2}, y=4y=x2,y=4

Find the integral. - ∫(5x−3)2dx\int(5 x-3)^{2} d x∫(5x−3)2dx

Find the integral. - ∫18−xdx\int \frac{1}{8-x} d x∫8−x1​dx

Find the integral. - ∫3ez8zdz\int \frac{3 \mathrm{e}^{\sqrt{\mathrm{z}}}}{8 \sqrt{\mathrm{z}}} \mathrm{dz}∫8z​3ez​​dz