8 - Applications of Integration

Chapter 8

Applications of Integration

Volumes of Solids of Revolution

Solids of Revolution

Disk Method

Washer Method(1)

Cylindrical Shell Method(2)

x-axis

Rotation about:

y-axis

2 or

2

2 or

2

Difference of Shells Method(2)(3)

2 or

2

2 or

2

Area Cross Section Method(4)

Notes:

1. The Washer Method is an extension of the Disk Method.

2. is the radius of the cylindrical shell. In cases where there is a gap between the axis of revolution and the functions being revolved, is the distance between the axis of revolution and either or , as appropriate.

3. The Difference of Shells Method is an extension of the Cylindrical Shell Method.

4. The function is the area of the cross section being integrated.

Chapter 8

Applications of Integration

Disk and Washer Methods

The formulas for the Disk Method and Washer Method for calculating volumes of revolution are provided above. Below, we present an approach that can be used to calculate volumes of revolution using these methods.

Under the Disk Method, we integrate the area of the region between a curve and its axis of

revolution to obtain volume. Since each cross-section of the resulting object will be a circle, we

use the formula

as our starting point. The resulting formula is:

or

The Washer Method is simply a dual application of the Disk Method. Consider the illustration at right. If we want the area of the shaded region, we subtract the area of the smaller circle from the area of the larger circle. The same occurs with the Washer Method; since we integrate cross-sectional area to find volume, so to obtain the volume of revolution of a region between the two curves we integrate the difference in the areas between the two curves.

Below is a set of steps that can be used to determine the volume of revolution of a region between two curves. The approach is illustrated based on the following example:

Example: Find the volume that results from revolving the region between the curves 2

and

about the line 6.

Steps

1. Graph the equations provided and any other information given in the problem (illustrated below). Then, isolate the section of the graph that we want to work with (illustrated at right). The disks we will use are shown as green and orange vertical lines. The dashed objects are reflections of the curves and disks over the axis of revolution; these give us an idea of what the central cross-section of the 3 shape will look like after revolution. You

do not need to draw these.

Integration Interval

Chapter 8

Applications of Integration

2. Identify whether there is a gap between the region to be revolved and the axis of revolution. In the example, the axis of revolution is 6, so there is clearly a gap between a) the red and blue curves, and b) the axis of revolution. Therefore, we will use the Washer Method.

3. Set up the integral form to be used.

a. Disk Method:

radius

or

radius

b. Washer Method:

big radius small radius

or

big radius small radius

4. Identify the variable of integration (i.e., are we using or ?). The disks used must be perpendicular to the axis of revolution.

a. If we are revolving around an axis, use the variable of that axis.

b. If the axis of revolution is a line of the form,

or

, use the opposite

variable from the one that occurs in the equation of the axis. In the example, the

axis of revolution is 6, so we will integrate with respect to .

Note: The expressions used in the integration must be in terms of the variable of

integration. So, for example, if the variable of integration is and the equation of a

curve is given as

, we must invert this to the form

before

integrating.

5. Identify the limits of integration. In the example, the curves intersect at 4. This results in an equation for volume in the form:

0 and

big radius small radius

6. Substitute the expressions for the big and small radii inside the integral. In the example, we have the following:

a. big radius 6 b. small radius 6 2

This results in the following:

~ .

Note that this matches the value calculated using the Difference of Shells Method below.

Chapter 8

Applications of Integration

Cylindrical Shell Methods

The formulas for the Cylindrical Shell Method and Difference of Shells Method for calculating volumes of revolution are provided above. Below, we present an approach that can be used to calculate volumes of revolution using these methods.

Under the Cylindrical Shell Method, we integrate the volume of a shell across the appropriate values of or . We use the formula for the volume of a cylinder as our starting point (i.e.,

2 , where is typically the function provided). The resulting formula is:

2

or

2

The Difference of Shells Method is essentially a dual application of the Cylindrical Shell Method. We want the volume of the cylinder whose height is the difference between two functions (see illustration at right).

Below is a set of steps that can be used to determine the volume of revolution of a region between two curves. The approach is illustrated based on the following example:

Example: Find the volume that results from revolving the region between the curves 2

and

about the line 6.

Steps

1. Graph the equations provided and any other information given in the problem (illustrated below left). Then, isolate the section of the graph that we want to work with (illustrated below right). Also shown are reflections of the curves over the axis of revolution (dashed curves); this allows us to see the "other side" of the cylindrical shells we will use. A typical shell is shown as a green cylinder.

Integration Interval

Chapter 8

Applications of Integration

2. Identify whether the integration involves one or two curves. a. One curve: Use the Cylindrical Shell Method. b. Two curves: Use the Difference of Shells Method. This is the case in the example.

3. Set up the integral form to be used. Let be the radius of the shell.

a. Cylindrical Shell Method: 2

or 2

.

b. Difference of Shells Method: 2

difference of shell heights or

2

difference of shell heights .

4. Identify the variable of integration (i.e., are we using or ?). The shells used must be parallel to the axis of revolution.

a. If we are revolving around an axis, use the opposite variable of that axis.

b. If the axis of revolution is a line of the form,

or

, use the same

variable as the one that occurs in the equation of the axis. In the example, the

axis of revolution is 6, so we will integrate with respect to .

2

difference of shell heights

5. Identify the limits of integration. In the example, the curves intersect at 4. This results in an equation for volume in the form:

2

difference of shell heights

0 and

6. Substitute the expressions for and the difference of shell heights into the integral. In

the example, we need to convert each equation to the form

because is the

variable of integration:

a.

so

2

2 so

The difference of shell heights, then, is 2

1 4

2.

b. The radius of a shell is the difference between the line in the interval, so the radius is 6 .

This results in the following:

6 and the value of

~ .

Note that this matches the value calculated using the Washer Method above.

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