Question 1

Evaluate the integral. $\int_{0}^{\pi / 4} 3(1+\tan t)^{3} \sec ^{2} t~~ d t$

Accepted Answer

The answer of Evaluate the integral. \(\int_{0}^{\pi / 4} 3(1+\tan...

Question 2

Approximate the area under the curve $y=\sin x$ from $0 \text { to } \frac{\pi}{2}$ using ten approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Accepted Answer

A)  0.36 
B)  0.02 
C)  0.72 
D)  0.98 
E)  1.08 
A)  0.36 
B)  0.02 
C)  0.72 
D)  0.98 
E)  1.08

Question 3

Determine a region whose area is equal to $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \tan \frac{i \pi}{2 n}$ .

Accepted Answer

A)  $y=\tan x, 0 \leq x \leq \frac{\pi}{3}$ 
B)  $y=\tan x, 0 \leq x \leq \frac{\pi}{5}$ 
C)  $y=\tan x, 0 \leq x \leq \frac{\pi}{6}$ 
D)  $y=\tan x, 0 \leq x \leq \frac{\pi}{4}$ 
E)  $y=\tan x, 0 \leq x \leq \frac{\pi}{2}$ 
A)  $y=\tan x, 0 \leq x \leq \frac{\pi}{3}$ 
B)  $y=\tan x, 0 \leq x \leq \frac{\pi}{5}$ 
C)  $y=\tan x, 0 \leq x \leq \frac{\pi}{6}$ 
D)  $y=\tan x, 0 \leq x \leq \frac{\pi}{4}$ 
E)  $y=\tan x, 0 \leq x \leq \frac{\pi}{2}$

Question 4

Evaluate the integral if it exists. $\int\left(\frac{5-x}{x}\right)^{2} d x$

Accepted Answer

A)  $1-10 \ln x+C$ 
B)  $x-10 \ln x-\frac{25}{x}+C$ 
C)  $\ln x-25 x+C$ 
D)  $x-\frac{1}{10 \ln x}+C$ 
E)  none of these 
A)  $1-10 \ln x+C$ 
B)  $x-10 \ln x-\frac{25}{x}+C$ 
C)  $\ln x-25 x+C$ 
D)  $x-\frac{1}{10 \ln x}+C$ 
E)  none of these

Question 5

Evaluate the integral by making the given substitution. $\int \cos 5 x d x, u=5 x$

Accepted Answer

The answer of Evaluate the integral by making the given...

Question 6

If $h^{\prime}$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 2 and 5 ?

Accepted Answer

A)  $\int_{2}^{5} h^{t}(t) d t$ 
B)  $h^{\prime}(5)-h^{\prime}(2)$ 
C)  $\int_{5}^{2} h(t) d t$ 
A)  $\int_{2}^{5} h^{t}(t) d t$ 
B)  $h^{\prime}(5)-h^{\prime}(2)$ 
C)  $\int_{5}^{2} h(t) d t$

Question 7

Use the given graph of $f$ to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

Accepted Answer

A)  8 
B)  6 
C)  4 
D)  3.5 
E)  4.5 
A)  8 
B)  6 
C)  4 
D)  3.5 
E)  4.5

Question 8

Evaluate the integral. $\int_{0}^{9}\left(6+6 y-y^{2}\right) d y$

Accepted Answer

A)  54 
B)  $34$ 
C)  $94$ 
D)  $84$ 
E)  $74$ 
A)  54 
B)  $34$ 
C)  $94$ 
D)  $84$ 
E)  $74$

Question 9

Evaluate the integral. $\int_{0}^{1 / 2} \frac{4 d r}{\sqrt{1-r^{2}}}$

Accepted Answer

A)  $\frac{\pi}{4}$ 
B)  $\frac{1}{5}$ 
C)  $\frac{\pi}{5}$ 
D)  $\frac{1}{4}$ 
E)  $\frac{2 \pi}{3}$ 
A)  $\frac{\pi}{4}$ 
B)  $\frac{1}{5}$ 
C)  $\frac{\pi}{5}$ 
D)  $\frac{1}{4}$ 
E)  $\frac{2 \pi}{3}$

Question 10

If $\int_{0}^{6} f(x) d x=15$ and $\int_{0}^{4} f(x) d x=6$ , find $\int_{4}^{6} f(x) d x$ .

Accepted Answer

A)  6 
B)  21 
C)  15 
D)  9 
E)  $-15$ 
A)  6 
B)  21 
C)  15 
D)  9 
E)  $-15$

Question 11

Evaluate by interpreting it in terms of areas. $\int_{-1}^{3}(6-x) ~ d x$

Accepted Answer

The answer of Evaluate by interpreting it in terms of...

Question 12

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{\sqrt{x}} \frac{2 z^{2}}{z^{4}+1} d z$

Accepted Answer

The answer of Use Part 1 of the Fundamental Theorem...

Question 13

Evaluate the Riemann sum for $f(r)=36-r^{2}, 0 \leq r \leq 2$ with four subintervals, taking the sample points to be right endpoints.

Accepted Answer

A)  $68.25$ 
B)  $69.75$ 
C)  $70.75$ 
D)  $70.25$ 
E)  $69.25$ 
A)  $68.25$ 
B)  $69.75$ 
C)  $70.75$ 
D)  $70.25$ 
E)  $69.25$

Question 14

Find the general indefinite integral. $\int \frac{\sin 120 t}{\sin 60 t} d t$

Accepted Answer

A)  $-\frac{\sin 120 t}{120}+C$ 
B)  $-\frac{\cos 120 t}{120}+C$ 
C)  $\frac{\sin 60 t}{30}+C$ 
D)  $\frac{\sin 120 t}{120}+C$ 
E)  $\frac{\cos 120 t}{30}+C$ 
A)  $-\frac{\sin 120 t}{120}+C$ 
B)  $-\frac{\cos 120 t}{120}+C$ 
C)  $\frac{\sin 60 t}{30}+C$ 
D)  $\frac{\sin 120 t}{120}+C$ 
E)  $\frac{\cos 120 t}{30}+C$

Question 15

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4}+1} d z$

Accepted Answer

A)  $\frac{\sqrt{x+1}}{x^{2}+2}$ 
B)  $\frac{\sqrt{x^{2}+1}}{2}$ 
C)  none of these 
D)  $\frac{\sqrt{x}}{x^{2}+1}$ 
E)  $\frac{\sqrt{x}}{2\left(x^{2}+1\right)}$ 
A)  $\frac{\sqrt{x+1}}{x^{2}+2}$ 
B)  $\frac{\sqrt{x^{2}+1}}{2}$ 
C)  none of these 
D)  $\frac{\sqrt{x}}{x^{2}+1}$ 
E)  $\frac{\sqrt{x}}{2\left(x^{2}+1\right)}$

Question 16

Find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $f(x)=\sqrt{\tan x}, 0 \leq x \leq \pi$

Accepted Answer

The answer of Find an expression for the area under...

Question 17

Evaluate the indefinite integral. $\int 7 e^{\cos x} \sin x d x$

Accepted Answer

A)  $-e^{\cos x} \sin x+C$ 
B)  $-7 e^{\cos x}+C$ 
C)  $e^{7 \sin x}+C$ 
D)  $7 e^{\cos x} \sin x+C$ 
E)  $-7 \sin \left(e^{\cos x}\right)+C$ 
A)  $-e^{\cos x} \sin x+C$ 
B)  $-7 e^{\cos x}+C$ 
C)  $e^{7 \sin x}+C$ 
D)  $7 e^{\cos x} \sin x+C$ 
E)  $-7 \sin \left(e^{\cos x}\right)+C$

Question 18

Find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $f(x)=x^{2}+\sqrt{1+2 x}, \quad 4 \leq x \leq 7$

Accepted Answer

The answer of Find an expression for the area under...

Question 19

Evaluate the integral. $\int_{1}^{4} \frac{x^{2}+6}{\sqrt{x}} d x$

Accepted Answer

A)  $24.4$ 
B)  $34.4$ 
C)  $39.4$ 
D)  $29.4$ 
E)  $44.4$ 
A)  $24.4$ 
B)  $34.4$ 
C)  $39.4$ 
D)  $29.4$ 
E)  $44.4$

Question 20

Evaluate the definite integral. $\int_{e^{84}}^{e^{25}} \frac{d x}{x \sqrt{\ln x}}$

Accepted Answer

The answer of Evaluate the definite integral. \(\int_{e^{84}}^{e^{25}} \frac{d x}{x...

Evaluate the integral. $\int_{0}^{\pi / 4} 3(1+\tan t)^{3} \sec ^{2} t~~ d t$

Approximate the area under the curve $y=\sin x$ from $0 \text { to } \frac{\pi}{2}$ using ten approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Determine a region whose area is equal to $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \tan \frac{i \pi}{2 n}$ .

Evaluate the integral if it exists. $\int\left(\frac{5-x}{x}\right)^{2} d x$

Evaluate the integral by making the given substitution. $\int \cos 5 x d x, u=5 x$

If $h^{\prime}$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 2 and 5 ?

Use the given graph of $f$ to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

Evaluate the integral. $\int_{0}^{9}\left(6+6 y-y^{2}\right) d y$

Evaluate the integral. $\int_{0}^{1 / 2} \frac{4 d r}{\sqrt{1-r^{2}}}$

If $\int_{0}^{6} f(x) d x=15$ and $\int_{0}^{4} f(x) d x=6$ , find $\int_{4}^{6} f(x) d x$ .

Evaluate by interpreting it in terms of areas. $\int_{-1}^{3}(6-x) ~ d x$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{\sqrt{x}} \frac{2 z^{2}}{z^{4}+1} d z$

Evaluate the Riemann sum for $f(r)=36-r^{2}, 0 \leq r \leq 2$ with four subintervals, taking the sample points to be right endpoints.

Find the general indefinite integral. $\int \frac{\sin 120 t}{\sin 60 t} d t$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4}+1} d z$

Find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $f(x)=\sqrt{\tan x}, 0 \leq x \leq \pi$

Evaluate the indefinite integral. $\int 7 e^{\cos x} \sin x d x$

Find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $f(x)=x^{2}+\sqrt{1+2 x}, \quad 4 \leq x \leq 7$

Evaluate the integral. $\int_{1}^{4} \frac{x^{2}+6}{\sqrt{x}} d x$

Evaluate the definite integral. $\int_{e^{84}}^{e^{25}} \frac{d x}{x \sqrt{\ln x}}$

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. ∫0π/43(1+tan⁡t)3sec⁡2t dt\int_{0}^{\pi / 4} 3(1+\tan t)^{3} \sec ^{2} t~~ d t∫0π/4​3(1+tant)3sec2t dt

Approximate the area under the curve y=sin⁡xy=\sin xy=sinx from 0 to π20 \text { to } \frac{\pi}{2}0 to 2π​ using ten approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Determine a region whose area is equal to lim⁡n→∞∑i=1nπ2ntan⁡iπ2n\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \tan \frac{i \pi}{2 n}limn→∞​∑i=1n​2nπ​tan2niπ​ .

Evaluate the integral if it exists. ∫(5−xx)2dx\int\left(\frac{5-x}{x}\right)^{2} d x∫(x5−x​)2dx

Evaluate the integral by making the given substitution. ∫cos⁡5xdx,u=5x\int \cos 5 x d x, u=5 x∫cos5xdx,u=5x

If h′h^{\prime}h′ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 2 and 5 ?

Use the given graph of fff to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

Evaluate the integral. ∫09(6+6y−y2)dy\int_{0}^{9}\left(6+6 y-y^{2}\right) d y∫09​(6+6y−y2)dy

Evaluate the integral. ∫01/24dr1−r2\int_{0}^{1 / 2} \frac{4 d r}{\sqrt{1-r^{2}}}∫01/2​1−r2​4dr​

If ∫06f(x)dx=15\int_{0}^{6} f(x) d x=15∫06​f(x)dx=15 and ∫04f(x)dx=6\int_{0}^{4} f(x) d x=6∫04​f(x)dx=6 , find ∫46f(x)dx\int_{4}^{6} f(x) d x∫46​f(x)dx .

Evaluate by interpreting it in terms of areas. ∫−13(6−x) dx\int_{-1}^{3}(6-x) ~ d x∫−13​(6−x) dx

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=∫1x2z2z4+1dzh(x)=\int_{1}^{\sqrt{x}} \frac{2 z^{2}}{z^{4}+1} d zh(x)=∫1x​​z4+12z2​dz

Evaluate the Riemann sum for f(r)=36−r2,0≤r≤2f(r)=36-r^{2}, 0 \leq r \leq 2f(r)=36−r2,0≤r≤2 with four subintervals, taking the sample points to be right endpoints.

Find the general indefinite integral. ∫sin⁡120tsin⁡60tdt\int \frac{\sin 120 t}{\sin 60 t} d t∫sin60tsin120t​dt

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=∫1xz2z4+1dzh(x)=\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4}+1} d zh(x)=∫1x​​z4+1z2​dz

Find an expression for the area under the graph of fff as a limit. Do not evaluate the limit. f(x)=tan⁡x,0≤x≤πf(x)=\sqrt{\tan x}, 0 \leq x \leq \pif(x)=tanx​,0≤x≤π

Evaluate the indefinite integral. ∫7ecos⁡xsin⁡xdx\int 7 e^{\cos x} \sin x d x∫7ecosxsinxdx

Find an expression for the area under the graph of fff as a limit. Do not evaluate the limit. f(x)=x2+1+2x,4≤x≤7f(x)=x^{2}+\sqrt{1+2 x}, \quad 4 \leq x \leq 7f(x)=x2+1+2x​,4≤x≤7

Evaluate the integral. ∫14x2+6xdx\int_{1}^{4} \frac{x^{2}+6}{\sqrt{x}} d x∫14​x​x2+6​dx

Evaluate the definite integral. ∫e84e25dxxln⁡x\int_{e^{84}}^{e^{25}} \frac{d x}{x \sqrt{\ln x}}∫e84e25​xlnx​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. $\int_{0}^{\pi / 4} 3(1+\tan t)^{3} \sec ^{2} t~~ d t$

Approximate the area under the curve $y=\sin x$ from $0 \text { to } \frac{\pi}{2}$ using ten approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Determine a region whose area is equal to $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \tan \frac{i \pi}{2 n}$ .

Evaluate the integral if it exists. $\int\left(\frac{5-x}{x}\right)^{2} d x$

Evaluate the integral by making the given substitution. $\int \cos 5 x d x, u=5 x$

If $h^{\prime}$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 2 and 5 ?

Use the given graph of $f$ to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

Evaluate the integral. $\int_{0}^{9}\left(6+6 y-y^{2}\right) d y$

Evaluate the integral. $\int_{0}^{1 / 2} \frac{4 d r}{\sqrt{1-r^{2}}}$

If $\int_{0}^{6} f(x) d x=15$ and $\int_{0}^{4} f(x) d x=6$ , find $\int_{4}^{6} f(x) d x$ .

Evaluate by interpreting it in terms of areas. $\int_{-1}^{3}(6-x) ~ d x$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{\sqrt{x}} \frac{2 z^{2}}{z^{4}+1} d z$

Evaluate the Riemann sum for $f(r)=36-r^{2}, 0 \leq r \leq 2$ with four subintervals, taking the sample points to be right endpoints.

Find the general indefinite integral. $\int \frac{\sin 120 t}{\sin 60 t} d t$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4}+1} d z$

Find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $f(x)=\sqrt{\tan x}, 0 \leq x \leq \pi$

Evaluate the indefinite integral. $\int 7 e^{\cos x} \sin x d x$

Find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $f(x)=x^{2}+\sqrt{1+2 x}, \quad 4 \leq x \leq 7$

Evaluate the integral. $\int_{1}^{4} \frac{x^{2}+6}{\sqrt{x}} d x$

Evaluate the definite integral. $\int_{e^{84}}^{e^{25}} \frac{d x}{x \sqrt{\ln x}}$