Question 1

Consider the region between the graph of  f(x) =x−1f ( x )  = \sqrt { x - 1 }f(x) =x−1​  and the x-axis on the interval [1,5].
-Using four disks or washers approximate the volume of the solid that is obtained by revolving this region around the x-axis.

Accepted Answer

A)   6  π\piπ 
B)   7  π\piπ 
C)   8  π\piπ 
D)   9  π\piπE) All of the above
F) None of the above

Question 2

Consider the region between the graph of  f(x) =1−x2f ( x )  = 1 - x ^ { 2 }f(x) =1−x2  and the line y = 2 on the interval [0, 1].
-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 1.

Accepted Answer

A)   1.16  π\piπ 
B)   2.16  π\piπ 
C)   1.96  π\piπ 
D)   2.67  π\piπE) A) and B)
F) None of the above

Question 3

Find the work required to pump all of the water out of the top of a tank and up to the ground level, given that the tank is an upright cylinder with radius 4 feet and height 8 feet, buried so that its top is 2 feet below the surface.

Accepted Answer

The answer of Find the work required to pump all...

Question 4

Find the exact value of the arc length of the function $f ( x ) = 3 \cos x$ on the interval [0, $\pi$ ] using a definite integral.

Accepted Answer

The answer of Find the exact value of the arc...

Question 5

Consider the region between the graph of  f(x) =x−1f ( x )  = \sqrt { x - 1 }f(x) =x−1​  and the x-axis on the interval [1, 5].
-Using four shells approximate the volume of the solid that is obtained by revolving this region around the vertical line x = 5.

Accepted Answer

A)   15.68  π\piπ 
B)   17.68  π\piπ 
C)   13.68  π\piπ 
D)   19.68  π\piπE) A) and B)
F) B) and C) B

Question 6

Consider the region between the graphs of  f(x) =x2f ( x )  = x ^ { 2 }f(x) =x2  and  g(x) =3xg ( x )  = 3 xg(x) =3x  on the interval [0, 3].
-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

Accepted Answer

A)   11.5  π\piπ 
B)   12.5  π\piπ 
C)   13.5  π\piπ 
D)   14.5  π\piπE) B) and C)
F) A) and B)

Question 7

Consider the region between the graphs of  f(x) =x2f ( x )  = x ^ { 2 }f(x) =x2  and  g(x) =3xg ( x )  = 3 xg(x) =3x  on the interval [0, 3].
-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 4.

Accepted Answer

A)   12.5  π\piπ 
B)   20.5  π\piπ 
C)   18.5  π\piπ 
D)   22.5  π\piπE) B) and D)
F) None of the above

Question 8

Consider the region between the graph of  f(x) =x−1f ( x )  = \sqrt { x - 1 }f(x) =x−1​  and the x-axis on the interval [1, 5].
-Using four shells approximate the volume of the solid that is obtained by revolving this region around the y-axis.

Accepted Answer

A)   32.15  π\piπ 
B)   30.15  π\piπ 
C)   36.15  π\piπ 
D)   41.15  π\piπE) None of the above
F) All of the above

Question 9

Find the exact value of the arc length of the function $f ( x ) = \ln ( \sin x )$ on the interval [ $\pi$ /4, $\pi$ /2] using a definite integral.

Accepted Answer

The answer of Find the exact value of the arc...

Question 10

Use definite integrals to find the area of the surface of revolution obtained by revolving  f(x) =cos⁡xf ( x )  = \cos xf(x) =cosx  around the x-axis on the interval [  π\piπ  /2, 3  π\piπ  /2].

Accepted Answer

A)   5.59  π\piπ 
B)   6.59  π\piπ 
C)   3.59  π\piπ 
D)   4.59  π\piπE) None of the above
F) B) and D)

Question 11

Find the work required to pump the upper 3 feet of the water out of the top of an upright conical tank with top radius 4 feet and height 6 feet.

Accepted Answer

2059.2 $\pi$ lb-ft.

Question 12

Find the hydrostatic force exerted on one of the long sides of a rectangular water tank that is 6 feet wide, 10 feet long, and 4 feet deep.

Accepted Answer

The answer of Find the hydrostatic force exerted on one...

Question 13

Use definite integrals to find the centroid of the region bounded by the graphs of $f ( x ) = 2 - x ^ { 2 }$ , $f ( x ) = x$ , and the y-axis.

Accepted Answer

The answer of Use definite integrals to find the centroid...

Question 14

Consider the region between the graph of  f(x) =1−x2f ( x )  = 1 - x ^ { 2 }f(x) =1−x2  and the line y = 2 on the interval [0, 1].
-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the horizontal line y = 2.

Accepted Answer

A)   1.87  π\piπ 
B)   2.87  π\piπ 
C)   3.87  π\piπ 
D)   0.87  π\piπE) B) and D)
F) None of the above

Question 15

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = x ^ { 3 } e ^ { - 2 y }$$

Accepted Answer

The answer of Use separation of variables to solve the...

Question 16

A cone-shaped water tank with top radius 4 feet and height 6 feet is on an 8-ft-high platform. Find the work done to fill this depot completely through an opening at the bottom of the tank if we pump the water from the ground level.

Accepted Answer

The answer of A cone-shaped water tank with top radius...

Question 17

Find the exact value of the arc length of the function $f ( x ) = 2 x ^ { 3 / 2 }$ on the interval [0, 2] using a definite integral.

Accepted Answer

6.06

Question 18

Find the exact value of the arc length of the function  f(x) =2x+3f ( x )  = 2 x + 3f(x) =2x+3  on the interval [-2, 3] using a definite integral.

Accepted Answer

A)    353 \sqrt { 5 }35c-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)    454 \sqrt { 5 }45c-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)    555 \sqrt { 5 }55c-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'/>​ 
D)    545 \sqrt { 4 }54c-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) B) and C)
F) All of the above

Question 19

Find the mass of a 20-inch rod whose cross section is a 2 × 2 inch square, with density × inches from the left end given by  ρ(x) =3.6+0.6x−0.03x2\rho ( x )  = 3.6 + 0.6 x - 0.03 x ^ { 2 }ρ(x) =3.6+0.6x−0.03x2  grams per cubic inch.

Accepted Answer

A)   440
B)   224
C)   446
D)   448 E) A) and B)
F) A) and C)

Question 20

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = \frac { y } { x ^ { 2 } + 1 }$ , $y ( 1 ) = 1$

Accepted Answer

The answer of Use separation of variables to solve the...

Exam 6: Applications of Integration

Consider the region between the graph of f(x) =x−1f ( x ) = \sqrt { x - 1 }f(x) =x−1​ and the x-axis on the interval [1,5]. -Using four disks or washers approximate the volume of the solid that is obtained by revolving this region around the x-axis.

Correct AnswerverifiedShow Answer

Consider the region between the graph of f(x) =1−x2f ( x ) = 1 - x ^ { 2 }f(x) =1−x2 and the line y = 2 on the interval [0, 1]. -Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 1.

Correct AnswerverifiedShow Answer

Find the work required to pump all of the water out of the top of a tank and up to the ground level, given that the tank is an upright cylinder with radius 4 feet and height 8 feet, buried so that its top is 2 feet below the surface.

Correct AnswerverifiedShow Answer

Find the exact value of the arc length of the function f(x)=3cos⁡xf ( x ) = 3 \cos xf(x)=3cosx on the interval [0, π\piπ ] using a definite integral.

Correct AnswerverifiedShow Answer

Consider the region between the graph of f(x) =x−1f ( x ) = \sqrt { x - 1 }f(x) =x−1​ and the x-axis on the interval [1, 5]. -Using four shells approximate the volume of the solid that is obtained by revolving this region around the vertical line x = 5.

Correct AnswerverifiedB

Consider the region between the graphs of f(x) =x2f ( x ) = x ^ { 2 }f(x) =x2 and g(x) =3xg ( x ) = 3 xg(x) =3x on the interval [0, 3]. -Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

Correct AnswerverifiedShow Answer

Consider the region between the graphs of f(x) =x2f ( x ) = x ^ { 2 }f(x) =x2 and g(x) =3xg ( x ) = 3 xg(x) =3x on the interval [0, 3]. -Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 4.

Correct AnswerverifiedShow Answer

Consider the region between the graph of f(x) =x−1f ( x ) = \sqrt { x - 1 }f(x) =x−1​ and the x-axis on the interval [1, 5]. -Using four shells approximate the volume of the solid that is obtained by revolving this region around the y-axis.

Correct AnswerverifiedShow Answer

Find the exact value of the arc length of the function f(x)=ln⁡(sin⁡x)f ( x ) = \ln ( \sin x )f(x)=ln(sinx) on the interval [ π\piπ /4, π\piπ /2] using a definite integral.

Correct AnswerverifiedShow Answer

Use definite integrals to find the area of the surface of revolution obtained by revolving f(x) =cos⁡xf ( x ) = \cos xf(x) =cosx around the x-axis on the interval [ π\piπ /2, 3 π\piπ /2].

Correct AnswerverifiedShow Answer

Find the work required to pump the upper 3 feet of the water out of the top of an upright conical tank with top radius 4 feet and height 6 feet.

Correct Answerverified2059.2 \(\pi\) lb-ft.

Find the hydrostatic force exerted on one of the long sides of a rectangular water tank that is 6 feet wide, 10 feet long, and 4 feet deep.

Correct AnswerverifiedShow Answer

Use definite integrals to find the centroid of the region bounded by the graphs of f(x)=2−x2f ( x ) = 2 - x ^ { 2 }f(x)=2−x2 , f(x)=xf ( x ) = xf(x)=x , and the y-axis.

Correct AnswerverifiedShow Answer

Consider the region between the graph of f(x) =1−x2f ( x ) = 1 - x ^ { 2 }f(x) =1−x2 and the line y = 2 on the interval [0, 1]. -Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the horizontal line y = 2.

Correct AnswerverifiedShow Answer

Use separation of variables to solve the differential equation: dydx=x3e−2y\frac { d y } { d x } = x ^ { 3 } e ^ { - 2 y }dxdy​=x3e−2y

Correct AnswerverifiedShow Answer

A cone-shaped water tank with top radius 4 feet and height 6 feet is on an 8-ft-high platform. Find the work done to fill this depot completely through an opening at the bottom of the tank if we pump the water from the ground level.

Correct AnswerverifiedShow Answer

Find the exact value of the arc length of the function f(x)=2x3/2f ( x ) = 2 x ^ { 3 / 2 }f(x)=2x3/2 on the interval [0, 2] using a definite integral.

Correct Answerverified6.06

Find the exact value of the arc length of the function f(x) =2x+3f ( x ) = 2 x + 3f(x) =2x+3 on the interval [-2, 3] using a definite integral.

Correct AnswerverifiedShow Answer

Find the mass of a 20-inch rod whose cross section is a 2 × 2 inch square, with density × inches from the left end given by ρ(x) =3.6+0.6x−0.03x2\rho ( x ) = 3.6 + 0.6 x - 0.03 x ^ { 2 }ρ(x) =3.6+0.6x−0.03x2 grams per cubic inch.

Correct AnswerverifiedShow Answer

Use separation of variables to solve the initial value problem: dydx=yx2+1\frac { d y } { d x } = \frac { y } { x ^ { 2 } + 1 }dxdy​=x2+1y​ , y(1)=1y ( 1 ) = 1y(1)=1

Correct AnswerverifiedShow Answer

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1,5]. -Using four disks or washers approximate the volume of the solid that is obtained by revolving this region around the x-axis.

Correct Answer
verified

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1]. -Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 1.

Correct Answer
verified

Correct Answer
verified

Find the exact value of the arc length of the function $f ( x ) = 3 \cos x$ on the interval [0, $\pi$ ] using a definite integral.

Correct Answer
verified

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1, 5]. -Using four shells approximate the volume of the solid that is obtained by revolving this region around the vertical line x = 5.

Correct Answer
verified
B

Consider the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = 3 x$ on the interval [0, 3]. -Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

Correct Answer
verified

Consider the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = 3 x$ on the interval [0, 3]. -Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 4.

Correct Answer
verified

Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1, 5]. -Using four shells approximate the volume of the solid that is obtained by revolving this region around the y-axis.

Correct Answer
verified

Find the exact value of the arc length of the function $f ( x ) = \ln ( \sin x )$ on the interval [ $\pi$ /4, $\pi$ /2] using a definite integral.

Correct Answer
verified

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = \cos x$ around the x-axis on the interval [ $\pi$ /2, 3 $\pi$ /2].

Correct Answer
verified

Correct Answer
verified
2059.2 \(\pi\) lb-ft.

Correct Answer
verified

Use definite integrals to find the centroid of the region bounded by the graphs of $f ( x ) = 2 - x ^ { 2 }$ , $f ( x ) = x$ , and the y-axis.

Correct Answer
verified

Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1]. -Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the horizontal line y = 2.

Correct Answer
verified

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = x ^ { 3 } e ^ { - 2 y }$

Correct Answer
verified

Correct Answer
verified

Find the exact value of the arc length of the function $f ( x ) = 2 x ^ { 3 / 2 }$ on the interval [0, 2] using a definite integral.

Correct Answer
verified
6.06

Find the exact value of the arc length of the function $f ( x ) = 2 x + 3$ on the interval [-2, 3] using a definite integral.

Correct Answer
verified

Find the mass of a 20-inch rod whose cross section is a 2 × 2 inch square, with density × inches from the left end given by $\rho ( x ) = 3.6 + 0.6 x - 0.03 x ^ { 2 }$ grams per cubic inch.

Correct Answer
verified

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = \frac { y } { x ^ { 2 } + 1 }$ , $y ( 1 ) = 1$

Correct Answer
verified