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The Washer Method for Volumes of Solids for AP Calculus

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

The volume of a solid (with a hole in the middle) generated by revolving a region bounded by 2 curves:

About the x-axis:

    ; where f (x) = outer radius & g (x) = inner radius.

About the y-axis:

    ; where p(y ) = outer radius & q(y) = inner radius.

About a line x = h:

About a line y = k:

Example 1

Using the Washer Method, find the volume of the solid generated by revolving the region bounded by y = x 3 and y = x in the first quadrant about the x-axis.

Step 1. Draw a sketch. See Figure 12.4-12.

The Washer Method

    To find the points of intersection, set x = x3 x 3x = 0 or x (x2 – 1)=0, or x = – 1, 0, 1. In the first quadrant x = 0, 1.

Step 2. Determine the outer and inner radii of a washer, whose outer radius = x; and inner radius = x3.

Step 3. Set up an integral.

Step 4. Evaluate the integral.

    Verify your result with a calculator.

Example 2

Using the Washer Method and a calculator, find the volume of the solid generated by revolving the region in Example 1 about the line y = 2.

Step 1. Draw a sketch. See Figure 12.4-13.

The Washer Method

Step 2. Determine the outer and inner radii of a washer. The outer radius = (2 – x3) and inner radius = (2 – x ).

Step 3. Set up an integral.

The Washer Method

Step 4. Evaluate the integral.

    and obtain .
    The volume of the solid is .

Example 3

Using the Washer Method and a calculator, find the volume of the solid generated by revolving the region bounded by y = x2 and x = y2 about the y-axis.

Step 1. Draw a sketch. See Figure 12.4-14.

    Intersection points: y = x2; x = y2 y =
    Set x2 = x4 = x4x = 0 x = 0 or x = 1
    x = 0, y = 0 (0, 0)
    x = 1, y = 1 (1, 1).

Step 2. Determine the outer and inner radii of a washer, with outer radius:

    x = and inner radius: x = y2.

Step 3. Set up an integral.

    .

Step 4. Evaluate the integral

    Enter and obtain .
        The volume of the solid is .

Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

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