Question 1

Evaluate the integral. $\int_{0}^{1} x^{2}\left(5+2 x^{3}\right)^{2} d x$

Accepted Answer

$\frac{109}{9}$

Question 2

Find the area of the region to three decimal places that lies under the given curve. $y=\sqrt{2 x+2}, \quad\quad0 \leq x \leq 1$

Accepted Answer

A)  $1.724$ 
B)  $1.874$ 
C)  $4.974$ 
D)  $1.769$ 
E)  $2.729$ 
A)  $1.724$ 
B)  $1.874$ 
C)  $4.974$ 
D)  $1.769$ 
E)  $2.729$ A

Question 3

A table of values of an increasing function $f(x)$ is shown. Use the table to find an upper estimate of $\int_{0}^{25} f(x) d x$ . $X$ $f(x)$ 0 -45 5 -37 10 -27 15 9 20 10 25 23

Accepted Answer

-110

Question 4

Evaluate the integral if it exists. $\int \frac{2 \cos (\ln x)}{x} d x$

Accepted Answer

The answer of Evaluate the integral if it exists. \(\int...

Question 5

Evaluate the indefinite integral. $\int \sqrt{4 x} \sin \left(5+x^{3 / 2}\right) d x$

Accepted Answer

The answer of Evaluate the indefinite integral. \(\int \sqrt{4 x}...

Question 6

Evaluate the indefinite integral. $\int \cos ^{6} x \sin x d x$

Accepted Answer

A)  $-\frac{1}{7} \sin ^{7} x+C$ 
B)  $\frac{1}{7} \sin ^{7} x+C$ 
C)  $\frac{1}{7} \cos ^{7} x+C$ 
D)  $-\frac{1}{7} \cos ^{7} x+C$ 
E)  $\frac{1}{7} \cos ^{7} x+C$. 
A)  $-\frac{1}{7} \sin ^{7} x+C$ 
B)  $\frac{1}{7} \sin ^{7} x+C$ 
C)  $\frac{1}{7} \cos ^{7} x+C$ 
D)  $-\frac{1}{7} \cos ^{7} x+C$ 
E)  $\frac{1}{7} \cos ^{7} x+C$.

Question 7

Find the area of the region that lies under the given curve. $y=\sin x, 0 \leq x \leq \frac{\pi}{2}$

Accepted Answer

The answer of Find the area of the region that...

Question 8

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find a lower estimate for the distance that she traveled during these three seconds. $t(\mathrm{~s})$ 0 $0.5$ $1.0$ $1.5$ $2.0$ $2.5$ $3.0$ $(\mathrm{ft} / \mathrm{s})$ 0 2.8
3.5
6.9
8.2
12.2
16.3

Accepted Answer

The answer of The speed of a runner increased steadily...

Question 9

By reading values from the given graph of $f$ , use five rectangles to find a lower estimate, to the nearest whole number, for the area from 0 to 10 under the given graph of $f$ .

Accepted Answer

The answer of By reading values from the given graph...

Question 10

The velocity function (in meters per second) is given for a particle moving along a line. Find the distance traveled by the particle during the given time interval. $v(t)=8 t-8,0 \leq t \leq 5$

Accepted Answer

A)  36 m 
B)  72 m 
C)  100 m 
D)  64 m 
E)  68 m 
A)  36 m 
B)  72 m 
C)  100 m 
D)  64 m 
E)  68 m

Question 11

Use the Midpoint Rule with n = 5 to approximate the integral. $\int_{0}^{10} 5 \sin \sqrt{q}~~ d q$ Round your answer to three decimal places.

Accepted Answer

A)  $36.909$ 
B)  $36.409$ 
C)  $31.409$ 
D)  $35.909$ 
E)  $37.709$ 
A)  $36.909$ 
B)  $36.409$ 
C)  $31.409$ 
D)  $35.909$ 
E)  $37.709$

Question 12

Find the area of the region that lies to the right of the y-axis and to the left of the parabola $x=2 y-y^{2}$ .

Accepted Answer

The answer of Find the area of the region that...

Question 13

Evaluate the indefinite integral. $\int \frac{6+6 x}{\sqrt{7+6 x+3 x^{2}}} d x$

Accepted Answer

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

Question 14

If $F(x)=\int_{1}^{x} f(t) ~d t$ where $f(t)=\int_{1}^{t^{2}} \frac{\sqrt{2+u^{2}}}{u}$ find $F^{\prime \prime}(2)$ .

Accepted Answer

A)  $3 \sqrt{2}$ 
B)  $6 \sqrt{2}$ 
C)  $6 \sqrt{3}$ 
D)  $\frac{3 \sqrt{2}}{2}$ 
E)  $3 \sqrt{3}$ 
A)  $3 \sqrt{2}$ 
B)  $6 \sqrt{2}$ 
C)  $6 \sqrt{3}$ 
D)  $\frac{3 \sqrt{2}}{2}$ 
E)  $3 \sqrt{3}$

Question 15

Evaluate the indefinite integral. $\int \frac{e^{x}}{e^{x}+5} d x$

Accepted Answer

A)  $-\frac{1}{2} \ln \left(e^{x}+5\right)+C$ 
B)  $\ln \left(e^{x}+5\right)+C$ 
C)  $\frac{1}{2} \ln \left(e^{x}+5\right)+C$ 
D)  $-\ln \left(e^{x}+5\right)+C$ 
E)  $\ln \left(e^{x}-5\right)+C$ 
A)  $-\frac{1}{2} \ln \left(e^{x}+5\right)+C$ 
B)  $\ln \left(e^{x}+5\right)+C$ 
C)  $\frac{1}{2} \ln \left(e^{x}+5\right)+C$ 
D)  $-\ln \left(e^{x}+5\right)+C$ 
E)  $\ln \left(e^{x}-5\right)+C$

Question 16

An animal population is increasing at a rate of $16+51 t$ per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

Accepted Answer

A)  $2248$ 
B)  $2288$ 
C)  $2338$ 
D)  $2258$ 
E)  $2238$ 
A)  $2248$ 
B)  $2288$ 
C)  $2338$ 
D)  $2258$ 
E)  $2238$

Question 17

An animal population is increasing at a rate of $32+36 t$ per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

Accepted Answer

The answer of An animal population is increasing at a...

Question 18

Given that $\int_{4}^{6} f(x) d x=\frac{5}{37}$ , find $\int_{4}^{6} f(x) ~ d x$ .

Accepted Answer

The answer of Given that $\int_{4}^{6} f(x) d x=\frac{5}{37}$ ,...

Question 19

Find the derivative of the function. $g(x)=\int_{4}^{x^{2}} 4 \sqrt{1+t^{2}} d t$

Accepted Answer

The answer of Find the derivative of the function. \(g(x)=\int_{4}^{x^{2}}...

Question 20

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

Accepted Answer

The answer of Evaluate the definite integral. \(\int_{-1}^{1} \frac{\sin x}{2+x^{2}}...

Evaluate the integral. $\int_{0}^{1} x^{2}\left(5+2 x^{3}\right)^{2} d x$

Find the area of the region to three decimal places that lies under the given curve. $y=\sqrt{2 x+2}, \quad\quad0 \leq x \leq 1$

A table of values of an increasing function $f(x)$ is shown. Use the table to find an upper estimate of $\int_{0}^{25} f(x) d x$ . $X$ $f(x)$ 0 -45 5 -37 10 -27 15 9 20 10 25 23

Evaluate the integral if it exists. $\int \frac{2 \cos (\ln x)}{x} d x$

Evaluate the indefinite integral. $\int \sqrt{4 x} \sin \left(5+x^{3 / 2}\right) d x$

Evaluate the indefinite integral. $\int \cos ^{6} x \sin x d x$

Find the area of the region that lies under the given curve. $y=\sin x, 0 \leq x \leq \frac{\pi}{2}$

By reading values from the given graph of $f$ , use five rectangles to find a lower estimate, to the nearest whole number, for the area from 0 to 10 under the given graph of $f$ .

The velocity function (in meters per second) is given for a particle moving along a line. Find the distance traveled by the particle during the given time interval. $v(t)=8 t-8,0 \leq t \leq 5$

Use the Midpoint Rule with n = 5 to approximate the integral. $\int_{0}^{10} 5 \sin \sqrt{q}~~ d q$ Round your answer to three decimal places.

Find the area of the region that lies to the right of the y-axis and to the left of the parabola $x=2 y-y^{2}$ .

Evaluate the indefinite integral. $\int \frac{6+6 x}{\sqrt{7+6 x+3 x^{2}}} d x$

If $F(x)=\int_{1}^{x} f(t) ~d t$ where $f(t)=\int_{1}^{t^{2}} \frac{\sqrt{2+u^{2}}}{u}$ find $F^{\prime \prime}(2)$ .

Evaluate the indefinite integral. $\int \frac{e^{x}}{e^{x}+5} d x$

An animal population is increasing at a rate of $16+51 t$ per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

An animal population is increasing at a rate of $32+36 t$ per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

Given that $\int_{4}^{6} f(x) d x=\frac{5}{37}$ , find $\int_{4}^{6} f(x) ~ d x$ .

Find the derivative of the function. $g(x)=\int_{4}^{x^{2}} 4 \sqrt{1+t^{2}} d t$

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

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 5: Applications of Integration

Evaluate the integral. ∫01x2(5+2x3)2dx\int_{0}^{1} x^{2}\left(5+2 x^{3}\right)^{2} d x∫01​x2(5+2x3)2dx

Find the area of the region to three decimal places that lies under the given curve. y=2x+2,0≤x≤1y=\sqrt{2 x+2}, \quad\quad0 \leq x \leq 1y=2x+2​,0≤x≤1

A table of values of an increasing function f(x)f(x)f(x) is shown. Use the table to find an upper estimate of ∫025f(x)dx\int_{0}^{25} f(x) d x∫025​f(x)dx . XXX f(x)f(x)f(x) 0 -45 5 -37 10 -27 15 9 20 10 25 23

Evaluate the integral if it exists. ∫2cos⁡(ln⁡x)xdx\int \frac{2 \cos (\ln x)}{x} d x∫x2cos(lnx)​dx

Evaluate the indefinite integral. ∫4xsin⁡(5+x3/2)dx\int \sqrt{4 x} \sin \left(5+x^{3 / 2}\right) d x∫4x​sin(5+x3/2)dx

Evaluate the indefinite integral. ∫cos⁡6xsin⁡xdx\int \cos ^{6} x \sin x d x∫cos6xsinxdx

Find the area of the region that lies under the given curve. y=sin⁡x,0≤x≤π2y=\sin x, 0 \leq x \leq \frac{\pi}{2}y=sinx,0≤x≤2π​

By reading values from the given graph of fff , use five rectangles to find a lower estimate, to the nearest whole number, for the area from 0 to 10 under the given graph of fff .

The velocity function (in meters per second) is given for a particle moving along a line. Find the distance traveled by the particle during the given time interval. v(t)=8t−8,0≤t≤5v(t)=8 t-8,0 \leq t \leq 5v(t)=8t−8,0≤t≤5

Use the Midpoint Rule with n = 5 to approximate the integral. ∫0105sin⁡q dq\int_{0}^{10} 5 \sin \sqrt{q}~~ d q∫010​5sinq​ dq Round your answer to three decimal places.

Find the area of the region that lies to the right of the y-axis and to the left of the parabola x=2y−y2x=2 y-y^{2}x=2y−y2 .

Evaluate the indefinite integral. ∫6+6x7+6x+3x2dx\int \frac{6+6 x}{\sqrt{7+6 x+3 x^{2}}} d x∫7+6x+3x2​6+6x​dx

If F(x)=∫1xf(t) dtF(x)=\int_{1}^{x} f(t) ~d tF(x)=∫1x​f(t) dt where f(t)=∫1t22+u2uf(t)=\int_{1}^{t^{2}} \frac{\sqrt{2+u^{2}}}{u}f(t)=∫1t2​u2+u2​​ find F′′(2)F^{\prime \prime}(2)F′′(2) .

Evaluate the indefinite integral. ∫exex+5dx\int \frac{e^{x}}{e^{x}+5} d x∫ex+5ex​dx

An animal population is increasing at a rate of 16+51t16+51 t16+51t per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

An animal population is increasing at a rate of 32+36t32+36 t32+36t per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

Given that ∫46f(x)dx=537\int_{4}^{6} f(x) d x=\frac{5}{37}∫46​f(x)dx=375​ , find ∫46f(x) dx\int_{4}^{6} f(x) ~ d x∫46​f(x) dx .

Find the derivative of the function. g(x)=∫4x241+t2dtg(x)=\int_{4}^{x^{2}} 4 \sqrt{1+t^{2}} d tg(x)=∫4x2​41+t2​dt

Evaluate the definite integral. ∫−11sin⁡x2+x2dx\int_{-1}^{1} \frac{\sin x}{2+x^{2}} d x∫−11​2+x2sinx​dx

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Evaluate the integral. $\int_{0}^{1} x^{2}\left(5+2 x^{3}\right)^{2} d x$

Find the area of the region to three decimal places that lies under the given curve. $y=\sqrt{2 x+2}, \quad\quad0 \leq x \leq 1$

A table of values of an increasing function $f(x)$ is shown. Use the table to find an upper estimate of $\int_{0}^{25} f(x) d x$ . $X$ $f(x)$ 0 -45 5 -37 10 -27 15 9 20 10 25 23

Evaluate the integral if it exists. $\int \frac{2 \cos (\ln x)}{x} d x$

Evaluate the indefinite integral. $\int \sqrt{4 x} \sin \left(5+x^{3 / 2}\right) d x$

Evaluate the indefinite integral. $\int \cos ^{6} x \sin x d x$

Find the area of the region that lies under the given curve. $y=\sin x, 0 \leq x \leq \frac{\pi}{2}$

By reading values from the given graph of $f$ , use five rectangles to find a lower estimate, to the nearest whole number, for the area from 0 to 10 under the given graph of $f$ .

The velocity function (in meters per second) is given for a particle moving along a line. Find the distance traveled by the particle during the given time interval. $v(t)=8 t-8,0 \leq t \leq 5$

Use the Midpoint Rule with n = 5 to approximate the integral. $\int_{0}^{10} 5 \sin \sqrt{q}~~ d q$ Round your answer to three decimal places.

Find the area of the region that lies to the right of the y-axis and to the left of the parabola $x=2 y-y^{2}$ .

Evaluate the indefinite integral. $\int \frac{6+6 x}{\sqrt{7+6 x+3 x^{2}}} d x$

If $F(x)=\int_{1}^{x} f(t) ~d t$ where $f(t)=\int_{1}^{t^{2}} \frac{\sqrt{2+u^{2}}}{u}$ find $F^{\prime \prime}(2)$ .

Evaluate the indefinite integral. $\int \frac{e^{x}}{e^{x}+5} d x$

An animal population is increasing at a rate of $16+51 t$ per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

An animal population is increasing at a rate of $32+36 t$ per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

Given that $\int_{4}^{6} f(x) d x=\frac{5}{37}$ , find $\int_{4}^{6} f(x) ~ d x$ .

Find the derivative of the function. $g(x)=\int_{4}^{x^{2}} 4 \sqrt{1+t^{2}} d t$

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