Question 1

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

Accepted Answer

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

Question 2

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = e ^ { - x }$ around the x-axis on the interval [-1, 2].

Accepted Answer

A)  6.33 $\pi$ 
B)  8.33 $\pi$ 
C)  9.33 $\pi$ 
D)  10.33 $\pi$ 
A)  6.33 $\pi$ 
B)  8.33 $\pi$ 
C)  9.33 $\pi$ 
D)  10.33 $\pi$

Question 3

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

Accepted Answer

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

Question 4

Find the hydrostatic force exerted on a dam in the shape of a trapezoid whose top is 320 feet long, whose base is 200 feet long, and whose height is 80 feet, given that the dam is completely full with water.

Accepted Answer

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

Question 5

Use separation of variables to solve the initial value problem: $\frac { d N } { d t } = 2 N$ , $N ( 1 ) = e ^ { 4 }$

Accepted Answer

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

Question 6

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 y-axis.

Accepted Answer

A)  33.27 $\pi$ 
B)  46.27 $\pi$ 
C)  53.27 $\pi$ 
D)  36.27 $\pi$ 
A)  33.27 $\pi$ 
B)  46.27 $\pi$ 
C)  53.27 $\pi$ 
D)  36.27 $\pi$

Question 7

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

Accepted Answer

A)  4.45 $\pi$ 
B)  8.85 $\pi$ 
C)  6.85 $\pi$ 
D)  7.65 $\pi$ 
A)  4.45 $\pi$ 
B)  8.85 $\pi$ 
C)  6.85 $\pi$ 
D)  7.65 $\pi$

Question 8

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 x-axis.

Accepted Answer

A)  2.47 $\pi$ 
B)  3.47 $\pi$ 
C)  1.47 $\pi$ 
D)  4.47 $\pi$ 
A)  2.47 $\pi$ 
B)  3.47 $\pi$ 
C)  1.47 $\pi$ 
D)  4.47 $\pi$

Question 9

Find the exact value of the arc length of the function $f ( x ) = 3 - x$ on the interval [1, 4] using a definite integral.

Accepted Answer

A)  $4 \sqrt { 2 }$ 
B)  $3 \sqrt { 2 }$ 
C)  $5 \sqrt { 2 }$ 
D)  $4 \sqrt { 3 }$ 
A)  $4 \sqrt { 2 }$ 
B)  $3 \sqrt { 2 }$ 
C)  $5 \sqrt { 2 }$ 
D)  $4 \sqrt { 3 }$

Question 10

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 horizontal line y = -1.

Accepted Answer

A)  20.07 $\pi$ 
B)  21.07 $\pi$ 
C)  23.07 $\pi$ 
D)  18.07 $\pi$ 
A)  20.07 $\pi$ 
B)  21.07 $\pi$ 
C)  23.07 $\pi$ 
D)  18.07 $\pi$

Question 11

Find the hydrostatic force exerted on a dam in the shape of an isosceles triangle whose top is 250 feet wide and whose total height is 100 feet, given that the dam is completely full with water.

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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 vertical line x = 5.

Accepted Answer

A)  15.07 $\pi$ 
B)  17.07 $\pi$ 
C)  14.07 $\pi$ 
D)  12.07 $\pi$ 
A)  15.07 $\pi$ 
B)  17.07 $\pi$ 
C)  14.07 $\pi$ 
D)  12.07 $\pi$

Question 14

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = \frac { x y } { x ^ { 2 } + 1 }$$

Accepted Answer

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

Question 15

Find the exact value of the arc length of the function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$ on the interval [-2, 2] using a definite integral.

Accepted Answer

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

Question 16

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 y-axis.

Accepted Answer

A)  $\pi$ 
B)  2 $\pi$ 
C)  1.5 $\pi$ 
D)  2.5 $\pi$ 
A)  $\pi$ 
B)  2 $\pi$ 
C)  1.5 $\pi$ 
D)  2.5 $\pi$

Question 17

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 y-axis.

Accepted Answer

A)  2 $\pi$ 
B)  2.5 $\pi$ 
C)  3 $\pi$ 
D)  1.5 $\pi$ 
A)  2 $\pi$ 
B)  2.5 $\pi$ 
C)  3 $\pi$ 
D)  1.5 $\pi$

Question 18

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 = 2.

Accepted Answer

A)  2.38 $\pi$ 
B)  2.83 $\pi$ 
C)  3.83 $\pi$ 
D)  4.83 $\pi$ 
A)  2.38 $\pi$ 
B)  2.83 $\pi$ 
C)  3.83 $\pi$ 
D)  4.83 $\pi$

Question 19

Use separation of variables to solve the differential equation: $$\frac { d y } { d x } = y ( 1 - y )$$

Accepted Answer

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

Question 20

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 horizontal line y = 4.

Accepted Answer

A)  20.33 $\pi$ 
B)  25.33 $\pi$ 
C)  29.33 $\pi$ 
D)  21.33 $\pi$ 
A)  20.33 $\pi$ 
B)  25.33 $\pi$ 
C)  29.33 $\pi$ 
D)  21.33 $\pi$

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

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = e ^ { - x }$ around the x-axis on the interval [-1, 2].

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

Find the hydrostatic force exerted on a dam in the shape of a trapezoid whose top is 320 feet long, whose base is 200 feet long, and whose height is 80 feet, given that the dam is completely full with water.

Use separation of variables to solve the initial value problem: $\frac { d N } { d t } = 2 N$ , $N ( 1 ) = e ^ { 4 }$

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 y-axis.

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

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 x-axis.

Find the exact value of the arc length of the function $f ( x ) = 3 - x$ on the interval [1, 4] using a definite integral.

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 horizontal line y = -1.

Find the hydrostatic force exerted on a dam in the shape of an isosceles triangle whose top is 250 feet wide and whose total height is 100 feet, given that the dam is completely full with water.

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

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 vertical line x = 5.

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = \frac { x y } { x ^ { 2 } + 1 }$

Find the exact value of the arc length of the function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$ on the interval [-2, 2] 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 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.

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 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 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 = 2.

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = y ( 1 - y )$

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 horizontal line y = 4.

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

Use separation of variables to solve the differential equation: dydx=(x4+2)2y\frac { d y } { d x } = \left( x ^ { 4 } + 2 \right) ^ { 2 } ydxdy​=(x4+2)2y

Use definite integrals to find the area of the surface of revolution obtained by revolving f(x)=e−xf ( x ) = e ^ { - x }f(x)=e−x around the x-axis on the interval [-1, 2].

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

Find the hydrostatic force exerted on a dam in the shape of a trapezoid whose top is 320 feet long, whose base is 200 feet long, and whose height is 80 feet, given that the dam is completely full with water.

Use separation of variables to solve the initial value problem: dNdt=2N\frac { d N } { d t } = 2 NdtdN​=2N , N(1)=e4N ( 1 ) = e ^ { 4 }N(1)=e4

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 y-axis.

Use definite integrals to find the area of the surface of revolution obtained by revolving f(x)=exf ( x ) = e ^ { x }f(x)=ex around the x-axis on the interval [-1, 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 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.

Find the exact value of the arc length of the function f(x)=3−xf ( x ) = 3 - xf(x)=3−x on the interval [1, 4] using a definite integral.

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 horizontal line y = -1.

Find the hydrostatic force exerted on a dam in the shape of an isosceles triangle whose top is 250 feet wide and whose total height is 100 feet, given that the dam is completely full with water.

Use separation of variables to solve the differential equation: dydx=x−2x3\frac { d y } { d x } = \frac { x - 2 } { \sqrt [ 3 ] { x } }dxdy​=3x​x−2​

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 vertical line x = 5.

Use separation of variables to solve the differential equation: dydx=xyx2+1\frac { d y } { d x } = \frac { x y } { x ^ { 2 } + 1 }dxdy​=x2+1xy​

Find the exact value of the arc length of the function f(x)=4−x2f ( x ) = \sqrt { 4 - x ^ { 2 } }f(x)=4−x2​ on the interval [-2, 2] 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 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.

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 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 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 = 2.

Use separation of variables to solve the differential equation: dydx=y(1−y)\frac { d y } { d x } = y ( 1 - y )dxdy​=y(1−y)

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 horizontal line y = 4.

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

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

Use definite integrals to find the area of the surface of revolution obtained by revolving $f ( x ) = e ^ { - x }$ around the x-axis on the interval [-1, 2].

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

Use separation of variables to solve the initial value problem: $\frac { d N } { d t } = 2 N$ , $N ( 1 ) = e ^ { 4 }$

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 y-axis.

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

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 x-axis.

Find the exact value of the arc length of the function $f ( x ) = 3 - x$ on the interval [1, 4] using a definite integral.

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 horizontal line y = -1.

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

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 vertical line x = 5.

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = \frac { x y } { x ^ { 2 } + 1 }$

Find the exact value of the arc length of the function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$ on the interval [-2, 2] 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 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.

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 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 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 = 2.

Use separation of variables to solve the differential equation: $\frac { d y } { d x } = y ( 1 - y )$

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 horizontal line y = 4.