Question 1

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

Accepted Answer

A)    1−10ln⁡x+C1-10 \ln x+C1−10lnx+C 
B)    x−10ln⁡x−25x+Cx-10 \ln x-\frac{25}{x}+Cx−10lnx−x25​+C 
C)    ln⁡x−25x+C\ln x-25 x+Clnx−25x+C 
D)    x−110ln⁡x+Cx-\frac{1}{10 \ln x}+Cx−10lnx1​+C 
E)   none of these F) B) and E)
G) A) and B)

Question 2

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 3

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 ?

Accepted Answer

A)    ∫25ht(t) dt\int_{2}^{5} h^{t}(t)  d t∫25​ht(t) dt 
B)    h′(5) −h′(2) h^{\prime}(5) -h^{\prime}(2) h′(5) −h′(2)  
C)    ∫52h(t) dt\int_{5}^{2} h(t)  d t∫52​h(t) dtD) A) and B)
E) A) and C)

Question 4

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.

Accepted Answer

A)    68.2568.2568.25 
B)    69.7569.7569.75 
C)    70.7570.7570.75 
D)    70.2570.2570.25 
E)    69.2569.2569.25F) C) and E)
G) A) and E)

Question 5

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

Accepted Answer

A)    −ecos⁡xsin⁡x+C-e^{\cos x} \sin x+C−ecosxsinx+C 
B)    −7ecos⁡x+C-7 e^{\cos x}+C−7ecosx+C 
C)    e7sin⁡x+Ce^{7 \sin x}+Ce7sinx+C 
D)    7ecos⁡xsin⁡x+C7 e^{\cos x} \sin x+C7ecosxsinx+C 
E)    −7sin⁡(ecos⁡x) +C-7 \sin \left(e^{\cos x}\right) +C−7sin(ecosx) +CF) A) and E)
G) A) and D)

Question 6

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 7

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

Accepted Answer

A)   54
B)    343434 
C)    949494 
D)    848484 
E)    747474F) D) and E)
G) A) and E)

Question 8

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

Accepted Answer

A)    π4\frac{\pi}{4}4π​ 
B)    15\frac{1}{5}51​ 
C)    π5\frac{\pi}{5}5π​ 
D)    14\frac{1}{4}41​ 
E)    2π3\frac{2 \pi}{3}32π​F) B) and C)
G) D) and E)

Question 9

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 10

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...

Question 11

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 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

Use the given graph of  fff  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 F) All of the above
G) A) and E)

Question 14

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 15

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

Accepted Answer

A)    −sin⁡120t120+C-\frac{\sin 120 t}{120}+C−120sin120t​+C 
B)    −cos⁡120t120+C-\frac{\cos 120 t}{120}+C−120cos120t​+C 
C)    sin⁡60t30+C\frac{\sin 60 t}{30}+C30sin60t​+C 
D)    sin⁡120t120+C\frac{\sin 120 t}{120}+C120sin120t​+C 
E)    cos⁡120t30+C\frac{\cos 120 t}{30}+C30cos120t​+CF) A) and B)
G) A) and C)

Question 16

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  .

Accepted Answer

A)   6
B)   21
C)   15
D)   9
E)    −15-15−15F) B) and C)
G) B) and E)

Question 17

Approximate the area under the curve  y=sin⁡xy=\sin xy=sinx  from  0 to π20 	ext { 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.

Accepted Answer

A)   0.36
B)   0.02
C)   0.72
D)   0.98
E)   1.08 F) B) and C)
G) A) and E)

Question 18

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

Accepted Answer

A)    y=tan⁡x,0≤x≤π3y=	an x, 0 \leq x \leq \frac{\pi}{3}y=tanx,0≤x≤3π​ 
B)    y=tan⁡x,0≤x≤π5y=	an x, 0 \leq x \leq \frac{\pi}{5}y=tanx,0≤x≤5π​ 
C)    y=tan⁡x,0≤x≤π6y=	an x, 0 \leq x \leq \frac{\pi}{6}y=tanx,0≤x≤6π​ 
D)    y=tan⁡x,0≤x≤π4y=	an x, 0 \leq x \leq \frac{\pi}{4}y=tanx,0≤x≤4π​ 
E)    y=tan⁡x,0≤x≤π2y=	an x, 0 \leq x \leq \frac{\pi}{2}y=tanx,0≤x≤2π​F) A) and E)
G) None of the above

Question 19

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

Accepted Answer

A)    x+1x2+2\frac{\sqrt{x+1}}{x^{2}+2}x2+2x+1c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
B)    x2+12\frac{\sqrt{x^{2}+1}}{2}2x2+1c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
C)   none of these
D)    xx2+1\frac{\sqrt{x}}{x^{2}+1}x2+1xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
E)    x2(x2+1) \frac{\sqrt{x}}{2\left(x^{2}+1\right) }2(x2+1) xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​F) A) and B)
G) A) and C)

Question 20

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

Accepted Answer

A)    24.424.424.4 
B)    34.434.434.4 
C)    39.439.439.4 
D)    29.429.429.4 
E)    44.444.444.4F) B) and C)
G) C) and D)

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

Correct AnswerverifiedShow Answer

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

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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 .

Correct AnswerverifiedShow Answer

Evaluate the definite integral. ∫e84e25dxxln⁡x\int_{e^{84}}^{e^{25}} \frac{d x}{x \sqrt{\ln x}}∫e84e25​xlnx​dx​

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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​ .

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

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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

Correct AnswerverifiedShow Answer

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

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

Correct AnswerverifiedShow Answer

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

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

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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

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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.

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

Correct AnswerverifiedShow Answer

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

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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

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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.

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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 ?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Correct Answer
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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.

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

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

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

Correct Answer
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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.

Correct Answer
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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 ?

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

Correct Answer
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