Question 1

Use the definite integral to find the area between the $x$-axis and the graph of $f(x)$ over the indicated interval.
-$f(x)=2 x+7 ;[1,5]$

Accepted Answer

A)  52 
B)  9 
C)  26 
D)  18 
A)  52 
B)  9 
C)  26 
D)  18

Question 2

Find the integral.
-$\int\left(\frac{4}{x}+2 e^{x}\right) d x$

Accepted Answer

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

Question 3

Use the definite integral to find the area between the $x$-axis and the graph of $f(x)$ over the indicated interval.
-$f(x)=x^{2}-6 x+9 ;[2,4]$

Accepted Answer

A)  $\frac{1}{3}$ 
B)  $\frac{7}{3}$ 
C)  $\frac{2}{3}$ 
D)  $\frac{4}{3}$ 
A)  $\frac{1}{3}$ 
B)  $\frac{7}{3}$ 
C)  $\frac{2}{3}$ 
D)  $\frac{4}{3}$

Question 4

Find the given indefinite integral.
-$\int \ln (7-2 x) d x$

Accepted Answer

A)  $\frac{1}{2}[(7-2 x) \ln (7-2 x)-14 x-1]+C$ 
B)  $-\frac{1}{2}[(7-2 x) \ln (7-2 x)-(7-2 x)]+C$ 
C)  $\left.\frac{1}{7}[\ln 7-2 x)-(7-2 x)\right]+C$ 
D)  $-\frac{1}{7}[(7-2 x) \ln (7-2 x)-(7-2 x)]+C$ 
A)  $\frac{1}{2}[(7-2 x) \ln (7-2 x)-14 x-1]+C$ 
B)  $-\frac{1}{2}[(7-2 x) \ln (7-2 x)-(7-2 x)]+C$ 
C)  $\left.\frac{1}{7}[\ln 7-2 x)-(7-2 x)\right]+C$ 
D)  $-\frac{1}{7}[(7-2 x) \ln (7-2 x)-(7-2 x)]+C$

Question 5

Solve the problem.
-Newton's law of cooling states that the rate at which a body changes temperature is proportional to the difference between its temperature and that of the surrounding medium. If a body is in air of temperature $25^{\circ}$ and the body cools from $110^{\circ}$ to $65^{\circ}$ in 30 minutes, find the temperature of the body after 60 minutes. (Round to nearest degree.)

Accepted Answer

A)  $47^{\circ}$ 
B)  $39^{\circ}$ 
C)  $44^{\circ}$ 
D)  $40^{\circ}$ 
A)  $47^{\circ}$ 
B)  $39^{\circ}$ 
C)  $44^{\circ}$ 
D)  $40^{\circ}$

Question 6

Solve the problem.
-The second derivative of a person's body temperature, with respect to the dosage of $x$ milligrams of a drug, is given by $D^{\prime}(x)=\frac{8}{x+5}$. One milligram raises the temperature $1.5^{\circ} \mathrm{C}$. Find the function $D(x)$ giving the total change in temperature as a function of $x$.

Accepted Answer

A)  $\mathrm{D}(\mathrm{x})=\ln \left|\frac{8}{\mathrm{x}+5}\right|+12.8$ 
B)  $\mathrm{D}(\mathrm{x})=8 \ln |\mathrm{x}+5|-12.8$ 
C)  $D(x)=8 \ln |x+5|+1.5$ 
D)  $\mathrm{D}(\mathrm{x})=\ln \left|\frac{8}{\mathrm{x}+5}\right|-1.5$ 
A)  $\mathrm{D}(\mathrm{x})=\ln \left|\frac{8}{\mathrm{x}+5}\right|+12.8$ 
B)  $\mathrm{D}(\mathrm{x})=8 \ln |\mathrm{x}+5|-12.8$ 
C)  $D(x)=8 \ln |x+5|+1.5$ 
D)  $\mathrm{D}(\mathrm{x})=\ln \left|\frac{8}{\mathrm{x}+5}\right|-1.5$

Question 7

Find the integral.
-$\int(1-6 x) e^{\left(3 x-9 x^{2}\right)} d x$

Accepted Answer

A)  $\frac{1}{3} e^{\left(3 x-9 x^{2}\right)}+C$ 
B)  $3(1-6 x) e^{\left(3 x-9 x^{2}\right)}+C$ 
C)  $\frac{1}{3}(1-6 x) e^{\left(3 x-9 x^{2}\right)}+C$ 
D)  $3 e^{\left(3 x-9 x^{2}\right)}+C$ 
A)  $\frac{1}{3} e^{\left(3 x-9 x^{2}\right)}+C$ 
B)  $3(1-6 x) e^{\left(3 x-9 x^{2}\right)}+C$ 
C)  $\frac{1}{3}(1-6 x) e^{\left(3 x-9 x^{2}\right)}+C$ 
D)  $3 e^{\left(3 x-9 x^{2}\right)}+C$

Question 8

Find the area bounded by the given curves.
-$y=x^{2}-5 x+4, y=-(x-1)^{2}$

Accepted Answer

A)  $\frac{7}{8}$ 
B)  $\frac{8}{7}$ 
C)  $\frac{8}{9}$ 
D)  $\frac{9}{8}$ 
A)  $\frac{7}{8}$ 
B)  $\frac{8}{7}$ 
C)  $\frac{8}{9}$ 
D)  $\frac{9}{8}$

Question 9

Find the integral.
-$\int \frac{8}{(y-9)^{3}} d y$

Accepted Answer

A)  $\frac{2}{(y-9)^{4}}+C$ 
B)  $\frac{-2}{(y-9)^{4}}+C$ 
C)  $\frac{4}{(y-9)^{2}}+C$ 
D)  $\frac{-4}{(y-9)^{2}}+C$ 
A)  $\frac{2}{(y-9)^{4}}+C$ 
B)  $\frac{-2}{(y-9)^{4}}+C$ 
C)  $\frac{4}{(y-9)^{2}}+C$ 
D)  $\frac{-4}{(y-9)^{2}}+C$

Question 10

Solve the problem.
-Suppose the supply function of a certain item is given by $S(x)=4 x+2$ and the demand function is $D(x)=14-x^{2}$. Find the producer's surplus.

Accepted Answer

A)  8 
B)  $\frac{8}{3}$ 
C)  16 
D)  $\frac{16}{3}$ 
A)  8 
B)  $\frac{8}{3}$ 
C)  16 
D)  $\frac{16}{3}$

Question 11

Provide an appropriate response.
-If $r(t)$ is the rate of change of revenue, then $\int_{a}^{b} r(t) d t$ is
i. the total revenue up to time b.
ii. the total revenue from time a to time $b$.
iii. the change of revenue at any moment.

Accepted Answer

A)  Both ii and iii could be correct. 
B)  Only i is correct. 
C)  Only iii is correct. 
D)  Only ii is correct. 
A)  Both ii and iii could be correct. 
B)  Only i is correct. 
C)  Only iii is correct. 
D)  Only ii is correct.

Question 12

Solve the problem.
-Suppose the supply function of a certain item is given by $S(x)=2 x+7$ and the demand function is $D(x)=27-x^{2 / 3}$. Find the producer's surplus.

Accepted Answer

A)  $\frac{32}{5}$ 
B)  32 
C)  64 
D)  $\frac{64}{5}$ 
A)  $\frac{32}{5}$ 
B)  32 
C)  64 
D)  $\frac{64}{5}$

Question 13

Solve the problem.
-The rate of infection of a disease (in people per month) is given by the function $f^{\prime}(t)=\frac{900 t}{t^{2}+1}$, where $t$ is the time in months since the disease broke out. Find the total number of infected people over the first 5 months of the disease.

Accepted Answer

A)  806 
B)  1466 
C)  2932 
D)  1613 
A)  806 
B)  1466 
C)  2932 
D)  1613

Question 14

Find the integral.
-$\int 2 e^{4 z} d z$

Accepted Answer

A)  $\frac{1}{2} e^{4 z}+C$ 
B)  $\frac{1}{4} e^{4 z}+C$ 
C)  $4 e^{4 z}+C$ 
D)  $2 e^{4 z}+C$ 
A)  $\frac{1}{2} e^{4 z}+C$ 
B)  $\frac{1}{4} e^{4 z}+C$ 
C)  $4 e^{4 z}+C$ 
D)  $2 e^{4 z}+C$

Question 15

Solve the problem.
-Find the producer's surplus if the supply function is given by $S(x)=x^{2}+4 x+20$. Assume that supply and demand are in equilibrium at $x=24$.

Accepted Answer

A)  10,638 
B)  10,368 
C)  10,386 
D)  10,836 
A)  10,638 
B)  10,368 
C)  10,386 
D)  10,836

Question 16

Find the particular solution of the differential equation.
-$\frac{d y}{d x}-x y-x=0 ; y=10$ when $x=1$

Accepted Answer

A)  $y=-\frac{1}{2}+\left(\frac{11}{2}\right) e^{x^{2}-1}$ 
B)  $\mathrm{y}=-\frac{1}{2}+11 \mathrm{e}^{-\mathrm{x}^{2}}$ 
C)  $y=-1+11 e^{-x^{2}}$ 
D)  $y=-1+11 e^{\left(x^{2}-1\right) / 2}$ 
A)  $y=-\frac{1}{2}+\left(\frac{11}{2}\right) e^{x^{2}-1}$ 
B)  $\mathrm{y}=-\frac{1}{2}+11 \mathrm{e}^{-\mathrm{x}^{2}}$ 
C)  $y=-1+11 e^{-x^{2}}$ 
D)  $y=-1+11 e^{\left(x^{2}-1\right) / 2}$

Question 17

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.
-$f(x)=x^{2}$ from $x=0$ to $x=4 ; n=4$

Accepted Answer

A)  36 
B)  33 
C)  27 
D)  30 
A)  36 
B)  33 
C)  27 
D)  30

Question 18

Use the definite integral to find the area between the $x$-axis and the graph of $f(x)$ over the indicated interval.
-$f(x)=x^{2}(x-2)^{2} ;[0,2]$

Accepted Answer

A)  $\frac{15}{16}$ 
B)  $\frac{16}{15}$ 
C)  $\frac{17}{15}$ 
D)  $\frac{15}{17}$ 
A)  $\frac{15}{16}$ 
B)  $\frac{16}{15}$ 
C)  $\frac{17}{15}$ 
D)  $\frac{15}{17}$

Question 19

Solve the problem.
-The rate of growth of profit (in millions) from an invention is approximated by $P^{\prime}(x)=x e^{-x^{2}}$, where $x$ represents time in years. The total profit in year 1 that the invention is in operation is $\$$ 45,000 . Find the total profit function $\mathrm{P}(\mathrm{x})$.

Accepted Answer

A)  $P(x)=-\frac{1}{2} e^{-x^{2}}-229,000$ 
B)  $P(x)=-.5 e^{-x^{2}}+0.229$ 
C)  $P(x)=-.5 e^{-x^{2}}+229,000$ 
D)  $P(x)=-\frac{1}{2} e^{-x^{2}}-0.229$ 
A)  $P(x)=-\frac{1}{2} e^{-x^{2}}-229,000$ 
B)  $P(x)=-.5 e^{-x^{2}}+0.229$ 
C)  $P(x)=-.5 e^{-x^{2}}+229,000$ 
D)  $P(x)=-\frac{1}{2} e^{-x^{2}}-0.229$

Question 20

Find the area of the shaded region.
-

Accepted Answer

A)  1.39 
B)  1.50 
C)  1.25 
D)  1.69 
A)  1.39 
B)  1.50 
C)  1.25 
D)  1.69

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x)=2 x+7 ;[1,5]$

Find the integral. - $\int\left(\frac{4}{x}+2 e^{x}\right) d x$

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x)=x^{2}-6 x+9 ;[2,4]$

Find the given indefinite integral. - $\int \ln (7-2 x) d x$

Find the integral. - $\int(1-6 x) e^{\left(3 x-9 x^{2}\right)} d x$

Find the area bounded by the given curves. - $y=x^{2}-5 x+4, y=-(x-1)^{2}$

Find the integral. - $\int \frac{8}{(y-9)^{3}} d y$

Solve the problem. -Suppose the supply function of a certain item is given by $S(x)=4 x+2$ and the demand function is $D(x)=14-x^{2}$ . Find the producer's surplus.

Provide an appropriate response. -If $r(t)$ is the rate of change of revenue, then $\int_{a}^{b} r(t) d t$ is i. the total revenue up to time b. ii. the total revenue from time a to time $b$ . iii. the change of revenue at any moment.

Solve the problem. -Suppose the supply function of a certain item is given by $S(x)=2 x+7$ and the demand function is $D(x)=27-x^{2 / 3}$ . Find the producer's surplus.

Solve the problem. -The rate of infection of a disease (in people per month) is given by the function $f^{\prime}(t)=\frac{900 t}{t^{2}+1}$ , where $t$ is the time in months since the disease broke out. Find the total number of infected people over the first 5 months of the disease.

Find the integral. - $\int 2 e^{4 z} d z$

Solve the problem. -Find the producer's surplus if the supply function is given by $S(x)=x^{2}+4 x+20$ . Assume that supply and demand are in equilibrium at $x=24$ .

Find the particular solution of the differential equation. - $\frac{d y}{d x}-x y-x=0 ; y=10$ when $x=1$

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. - $f(x)=x^{2}$ from $x=0$ to $x=4 ; n=4$

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x)=x^{2}(x-2)^{2} ;[0,2]$

Find the area of the shaded region. -

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Exam 13: Integral Calculus

Use the definite integral to find the area between the xxx -axis and the graph of f(x)f(x)f(x) over the indicated interval. - f(x)=2x+7;[1,5]f(x)=2 x+7 ;[1,5]f(x)=2x+7;[1,5]

Find the integral. - ∫(4x+2ex)dx\int\left(\frac{4}{x}+2 e^{x}\right) d x∫(x4​+2ex)dx

Use the definite integral to find the area between the xxx -axis and the graph of f(x)f(x)f(x) over the indicated interval. - f(x)=x2−6x+9;[2,4]f(x)=x^{2}-6 x+9 ;[2,4]f(x)=x2−6x+9;[2,4]

Find the given indefinite integral. - ∫ln⁡(7−2x)dx\int \ln (7-2 x) d x∫ln(7−2x)dx

Find the integral. - ∫(1−6x)e(3x−9x2)dx\int(1-6 x) e^{\left(3 x-9 x^{2}\right)} d x∫(1−6x)e(3x−9x2)dx

Find the area bounded by the given curves. - y=x2−5x+4,y=−(x−1)2y=x^{2}-5 x+4, y=-(x-1)^{2}y=x2−5x+4,y=−(x−1)2

Find the integral. - ∫8(y−9)3dy\int \frac{8}{(y-9)^{3}} d y∫(y−9)38​dy

Solve the problem. -Suppose the supply function of a certain item is given by S(x)=4x+2S(x)=4 x+2S(x)=4x+2 and the demand function is D(x)=14−x2D(x)=14-x^{2}D(x)=14−x2 . Find the producer's surplus.

Provide an appropriate response. -If r(t)r(t)r(t) is the rate of change of revenue, then ∫abr(t)dt\int_{a}^{b} r(t) d t∫ab​r(t)dt is i. the total revenue up to time b. ii. the total revenue from time a to time bbb . iii. the change of revenue at any moment.

Solve the problem. -Suppose the supply function of a certain item is given by S(x)=2x+7S(x)=2 x+7S(x)=2x+7 and the demand function is D(x)=27−x2/3D(x)=27-x^{2 / 3}D(x)=27−x2/3 . Find the producer's surplus.

Find the integral. - ∫2e4zdz\int 2 e^{4 z} d z∫2e4zdz

Solve the problem. -Find the producer's surplus if the supply function is given by S(x)=x2+4x+20S(x)=x^{2}+4 x+20S(x)=x2+4x+20 . Assume that supply and demand are in equilibrium at x=24x=24x=24 .

Find the particular solution of the differential equation. - dydx−xy−x=0;y=10\frac{d y}{d x}-x y-x=0 ; y=10dxdy​−xy−x=0;y=10 when x=1x=1x=1

Approximate the area under the curve and above the xxx -axis using nnn rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - f(x)=x2f(x)=x^{2}f(x)=x2 from x=0x=0x=0 to x=4;n=4x=4 ; n=4x=4;n=4

Use the definite integral to find the area between the xxx -axis and the graph of f(x)f(x)f(x) over the indicated interval. - f(x)=x2(x−2)2;[0,2]f(x)=x^{2}(x-2)^{2} ;[0,2]f(x)=x2(x−2)2;[0,2]