Exam 6: Applications of Integration

Consider the region between the graph of $f ( x ) = x ^ { 2 } + 1$ and the x-axis on the interval [0, 2]. -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.

(Multiple Choice)

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Question 41

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 = 2.

(Multiple Choice)

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Question 42

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = 3 \sqrt { y }$ , $y ( 3 ) = 4$

(Essay)

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Question 43

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = \frac { 1 } { x }$ around the x-axis on the interval [1, 4].

(Multiple Choice)

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Question 44

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = y ^ { 3 } \cos x$ , $y \left( \frac { \pi } { 6 } \right) = 1$

(Essay)

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Question 45

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 vertical line x = 1.

(Multiple Choice)

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Question 46

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 x-axis.

(Multiple Choice)

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Question 47

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }$ , $y ( 0 ) = 3$

(Essay)

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Question 48

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 x-axis.

(Multiple Choice)

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Question 49

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 x-axis.

(Multiple Choice)

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Question 50

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2]. -Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

(Multiple Choice)

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Question 51

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = \frac { 5 } { x y }$

(Essay)

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Question 52

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

(Essay)

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Question 53

Find the work required to pump all of the water out of the top of an upright cylindrical tank with radius 4 feet and height 8 feet.

(Short Answer)

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Question 54

Use definite integrals to find the centroid of the region between the graph of $f ( x ) = 2 \sqrt { x }$ and $g ( x ) = x$ on the interval [0, 4].

(Short Answer)

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Question 55

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = \frac { 3 } { x ^ { 2 } + 9 }$

(Essay)

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Question 56

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2]. -Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the x-axis.

(Multiple Choice)

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Question 57

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = \left( x ^ { 4 } + 2 \right) ^ { 2 }$

(Essay)

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Question 58

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 horizontal line y = -1.

(Multiple Choice)

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Question 59

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 horizontal line y = -1.

(Multiple Choice)

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Question 60

Consider the region between the graph of $f ( x ) = x ^ { 2 } + 1$ and the x-axis on the interval [0, 2]. -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 $f ( x ) = 1 - x ^ { 2 }$ 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 = 2.

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = 3 \sqrt { y }$ , $y ( 3 ) = 4$

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = \frac { 1 } { x }$ around the x-axis on the interval [1, 4].

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = y ^ { 3 } \cos x$ , $y \left( \frac { \pi } { 6 } \right) = 1$

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 vertical line x = 1.

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 x-axis.

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }$ , $y ( 0 ) = 3$

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 x-axis.

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 x-axis.

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2]. -Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = \frac { 5 } { x y }$

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

Find the work required to pump all of the water out of the top of an upright cylindrical tank with radius 4 feet and height 8 feet.

Use definite integrals to find the centroid of the region between the graph of $f ( x ) = 2 \sqrt { x }$ and $g ( x ) = x$ on the interval [0, 4].

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = \frac { 3 } { x ^ { 2 } + 9 }$

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2]. -Use the shell method to construct definite integrals to find 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: $\frac { d y } { d x } = \left( x ^ { 4 } + 2 \right) ^ { 2 }$

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 horizontal line y = -1.

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 horizontal line y = -1.

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

Consider the region between the graph of f(x)=x2+1f ( x ) = x ^ { 2 } + 1f(x)=x2+1 and the x-axis on the interval [0, 2]. -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)=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 = 2.

Use separation of variables to solve the initial value problem: dydx=3y\frac { d y } { d x } = 3 \sqrt { y }dxdy​=3y​ , y(3)=4y ( 3 ) = 4y(3)=4

Use definite integrals to find the area of the surface of revolution obtained by revolving f(x)=1xf ( x ) = \frac { 1 } { x }f(x)=x1​ around the x-axis on the interval [1, 4].

Use separation of variables to solve the initial value problem: dydx=y3cos⁡x\frac { d y } { d x } = y ^ { 3 } \cos xdxdy​=y3cosx , y(π6)=1y \left( \frac { \pi } { 6 } \right) = 1y(6π​)=1

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 vertical line x = 1.

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 x-axis.

Use separation of variables to solve the initial value problem: dydx=x21+x3\frac { d y } { d x } = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }dxdy​=1+x3x2​ , y(0)=3y ( 0 ) = 3y(0)=3

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 x-axis.

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 x-axis.

Consider the region between the graph of f(x)=−x+4f ( x ) = - x + 4f(x)=−x+4 and the x-axis on the interval [0, 2]. -Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

Use separation of variables to solve the differential equation: dydx=5xy\frac { d y } { d x } = \frac { 5 } { x y }dxdy​=xy5​

Use separation of variables to solve the differential equation: dydx=x3y2\frac { d y } { d x } = x ^ { 3 } y ^ { 2 }dxdy​=x3y2

Find the work required to pump all of the water out of the top of an upright cylindrical tank with radius 4 feet and height 8 feet.

Use definite integrals to find the centroid of the region between the graph of f(x)=2xf ( x ) = 2 \sqrt { x }f(x)=2x​ and g(x)=xg ( x ) = xg(x)=x on the interval [0, 4].

Use separation of variables to solve the differential equation: dydx=3x2+9\frac { d y } { d x } = \frac { 3 } { x ^ { 2 } + 9 }dxdy​=x2+93​

Consider the region between the graph of f(x)=−x+4f ( x ) = - x + 4f(x)=−x+4 and the x-axis on the interval [0, 2]. -Use the shell method to construct definite integrals to find 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=(x4+2)2\frac { d y } { d x } = \left( x ^ { 4 } + 2 \right) ^ { 2 }dxdy​=(x4+2)2

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 horizontal line y = -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 horizontal line y = -1.

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

Consider the region between the graph of $f ( x ) = x ^ { 2 } + 1$ and the x-axis on the interval [0, 2]. -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 $f ( x ) = 1 - x ^ { 2 }$ 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 = 2.

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = 3 \sqrt { y }$ , $y ( 3 ) = 4$

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = \frac { 1 } { x }$ around the x-axis on the interval [1, 4].

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = y ^ { 3 } \cos x$ , $y \left( \frac { \pi } { 6 } \right) = 1$

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 vertical line x = 1.

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 x-axis.

Use separation of variables to solve the initial value problem: $\frac { d y } { d x } = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }$ , $y ( 0 ) = 3$

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 x-axis.

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 x-axis.

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2]. -Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = \frac { 5 } { x y }$

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

Use definite integrals to find the centroid of the region between the graph of $f ( x ) = 2 \sqrt { x }$ and $g ( x ) = x$ on the interval [0, 4].

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = \frac { 3 } { x ^ { 2 } + 9 }$

Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2]. -Use the shell method to construct definite integrals to find 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: $\frac { d y } { d x } = \left( x ^ { 4 } + 2 \right) ^ { 2 }$

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 horizontal line y = -1.

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 horizontal line y = -1.