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.

- To find the points of intersection, set

*x*=

*x*

^{3}

*x*

^{3}–

*x*= 0 or

*x*(

*x*

^{2}– 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 = *x*^{3}.

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.

Step 2. Determine the outer and inner radii of a washer. The outer radius = (2 – *x*^{3}) and inner radius = (2 – *x* ).

Step 3. Set up an integral.

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* = *x*^{2} and *x* = y^{2} about the y-axis.

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

- Intersection points:

*y*=

*x*

^{2};

*x*=

*y*

^{2}

*y*=

- Set

*x*

^{2}=

*x*

^{4}=

*x*

^{4}–

*x*= 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*=

*y*

^{2}.

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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