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7.2 Disk Method (continued)

is bounded by the x-axis and the y-axis in the first quadrant and isrevolved about the y-axis. Find the volume of the solid formed.

1) Draw a sketch of the region to be rotated.Indicate a representative sliceand draw an arrowshowing the rotation.

2) Identify the radius

3) Write dV (expression for the volume)

4) Determine where the slices start and stop

5) Write the integral

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The region bounded by y= 9 - x2

, the x-axis, the y-axis is rotatedaround the y-axis. Find the volume of the solid formed.

Draw a sketch of the region to be rotated.

Indicate a representative slice and draw an arrowshowing the rotation.

Identify the length of the radius ____________

Write dV _____________

Determine where the slices start and stop ______

Write the integral _______________

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Find the volume of the solid generated by revolving the regionbounded by the graphs of the equations about the line y= 4.

Equations:

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Find the volume of the solid generated by revolving the regionbounded by the graphs of the equations about the line x= 6.

Equations:

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7.2 Finding Volume using the Washer Method

Example 1) Find the volume of the solid formed by revolving the

region bounded by the graphs y = !x and y = x2about the x-axis.

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This application of the method of slicing is called the washer method.The shape of the slice is a circle with a hole in it, so we subtract thearea of the inner circle from the area of the outer circle.

Washer Method Formula

OR

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Example 2) Find the volume of the solid enclosed by thecurves y = x and y = x2 when it is rotated about the y-axis.

What if the line was rotated about the line ?

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Example 3) Find the volume of the region bounded by thecurves y = x and y = x2when it is rotated about the line y = 3.

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If we use a horizontal slice, the disk now

has a hole in it, making it a washer.

The volume of the washer is:

outerradius

innerradius

thickness

Example 4) The region bounded by y = x2and y = 2x isrevolved about the y-axis. Find the volume.

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Example 5) The region to the right is enclosed

by x = 0, y = 0, x = 1 and y = x2+ 1.

What is the volume of the solid formed byrevolving this region about the x-axis?

What about if we revolved it around the y-axis?

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Example 7) Find the volume of the region bounded between x = 0,

y = 1, x = 1, y = (x-1)2+1 when it is rotated....

a) about the x-axis

b) about x = -1

c) about x = 2