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revolution mc-TY-volumes-2009-1 We sometimes need to calculate the volume of a solid which can be obtained by rotating a curve about the x-axis There is a 

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The 3-D model of the solid of revolution FORMULAS: V= ∫Adx , or respectively ∫Ady where A stands for the area of the typical disc and r=f(x) or r=f(y) depending on the axis of revolution

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www.rit.edu/asc Page 1 of 8

1. Finding volume of a solid of revolution using a disc method.

2. Finding volume of a solid of revolution using a washer method.

3. Finding volume of a solid of revolution using a shell method.

If a region in the plane is revolved about a given line, the resulting solid is a solid of revolution, and the

line is called the axis of revolution. When calculating the volume of a solid generated by revolving a

region bounded by a given function about an axis, follow the steps below:

1. Sketch the area and determine the axis of revolution, (this determines the variable of integration)

2. Sketch the cross-section, (disk, shell, washer) and determine the appropriate formula.

3. Determine the boundaries of the solid,

4. Set up the definite integral, and integrate.

1. Finding volume of a solid of revolution using a disc method.

The simplest solid of revolution is a right circular cylinder which is formed by revolving a rectangle about

an axis adjacent to one side of the rectangle, (the disc). To see how to calculate the volume of a general solid of revolution with a disc cross-section, using

integration techniques, consider the following solid of revolution formed by revolving the plane region

bounded by f(x), y-axis and the vertical line x=2 about the x-axis. (see Figure1 to 4 below):

Figure 1. The area under f(x), bounded by f(x), x-axis, Figure 2. Basic sketch of the solid of revolution

y-axis and the vertical line x=2 rotated about x-axis with few typical discs indicated. Figure 3. Family of discs Figure 4. The 3-D model of the solid of revolution.

Volumes by Integration

f(x) r=f(x)=y www.rit.edu/asc Page 2 of 8

FORMULAS: V=

Adx , or respectively Ady where A stands for the area of the typical disc.

Another words:

2rA and r=f(x) or r=f(y) depending on the axis of revolution.

1. The volume of the solid generated by a region under f(x) bounded by the x-axis and vertical lines

x=a and x=b, which is revolved about the x-axis is b a b a dxxfdxyV22S (disc with respect to x and r=y=f(x))

2. The volume of the solid generated by a region under f(y) (to the left of f(y) bounded by the y-axis,

and horizontal lines y=c and y=d which is revolved about the y-axis. d c d c dyyfdyxV22)(S (disc with respect to y and r=x=f(y)) Ex. 1. (Source: Paul Dawkins) http://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx Determine the volume of the solid generated by rotating the region bounded by

54)(2 xxxf

1x 4x and the x-axis about the x-axis.

Solution:

Step 1 is to sketch the bounding region and the solid obtained by rotating the region about the x-axis.

Here are both of these sketches.

Step 2: To get a cross section we cut the solid at any x, since the x-axis it the axis of rotation. f(x) c d r=f(y)=x www.rit.edu/asc Page 3 of 8

In this case the radius is simply the distance from the x-axis to the curve and this is nothing more than the

function value at that particular x as shown above. The cross-sectional area is the @22)(xfrxAS which in this case is equal to

25402685423422 xxxxxxxAS

Step3. Determine the boundaries which will represent the limits of integration. Working from left to right

the first cross section will occur at 1x , and the last cross section will occur at 4x . These are the limits of integration.

Step 4. Integrate to find the volume:

5

7825203

2625
1

254026854

4 1 2345
4 1 234
4 1 22
S SSS xxxxx dxxxxxdxxxdxxfdxxAV b a b a The volume of the solid generated by rotating the region bounded by

54)(2 xxxf

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