Question 1

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 2

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.

Accepted Answer

A)  15.68 $\pi$ 
B)  17.68 $\pi$ 
C)  13.68 $\pi$ 
D)  19.68 $\pi$ 
A)  15.68 $\pi$ 
B)  17.68 $\pi$ 
C)  13.68 $\pi$ 
D)  19.68 $\pi$ B

Question 3

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 4

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.

Accepted Answer

A)  1.16 $\pi$ 
B)  2.16 $\pi$ 
C)  1.96 $\pi$ 
D)  2.67 $\pi$ 
A)  1.16 $\pi$ 
B)  2.16 $\pi$ 
C)  1.96 $\pi$ 
D)  2.67 $\pi$

Question 5

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 6

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 7

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.

Accepted Answer

A)  $3 \sqrt { 5 }$ 
B)  $4 \sqrt { 5 }$ 
C)  $5 \sqrt { 5 }$ 
D)  $5 \sqrt { 4 }$ 
A)  $3 \sqrt { 5 }$ 
B)  $4 \sqrt { 5 }$ 
C)  $5 \sqrt { 5 }$ 
D)  $5 \sqrt { 4 }$

Question 8

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.

Accepted Answer

A)  12.5 $\pi$ 
B)  20.5 $\pi$ 
C)  18.5 $\pi$ 
D)  22.5 $\pi$ 
A)  12.5 $\pi$ 
B)  20.5 $\pi$ 
C)  18.5 $\pi$ 
D)  22.5 $\pi$

Question 9

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 10

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...

Question 11

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.

Accepted Answer

A)  32.15 $\pi$ 
B)  30.15 $\pi$ 
C)  36.15 $\pi$ 
D)  41.15 $\pi$ 
A)  32.15 $\pi$ 
B)  30.15 $\pi$ 
C)  36.15 $\pi$ 
D)  41.15 $\pi$

Question 12

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.

Accepted Answer

A)  440 
B)  224 
C)  446 
D)  448 
A)  440 
B)  224 
C)  446 
D)  448

Question 13

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.

Accepted Answer

A)  11.5 $\pi$ 
B)  12.5 $\pi$ 
C)  13.5 $\pi$ 
D)  14.5 $\pi$ 
A)  11.5 $\pi$ 
B)  12.5 $\pi$ 
C)  13.5 $\pi$ 
D)  14.5 $\pi$

Question 14

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 15

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.

Accepted Answer

A)  6 $\pi$ 
B)  7 $\pi$ 
C)  8 $\pi$ 
D)  9 $\pi$ 
A)  6 $\pi$ 
B)  7 $\pi$ 
C)  8 $\pi$ 
D)  9 $\pi$

Question 16

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 17

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 18

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 19

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.

Accepted Answer

A)  1.87 $\pi$ 
B)  2.87 $\pi$ 
C)  3.87 $\pi$ 
D)  0.87 $\pi$ 
A)  1.87 $\pi$ 
B)  2.87 $\pi$ 
C)  3.87 $\pi$ 
D)  0.87 $\pi$

Question 20

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].

Accepted Answer

A)  5.59 $\pi$ 
B)  6.59 $\pi$ 
C)  3.59 $\pi$ 
D)  4.59 $\pi$ 
A)  5.59 $\pi$ 
B)  6.59 $\pi$ 
C)  3.59 $\pi$ 
D)  4.59 $\pi$

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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

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.

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.

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.

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].

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 6: Applications of Integration

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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

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.

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.

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.

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].

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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

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.

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.

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.

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].