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

The volume of a solid of revolution using discs: Revolving about the *x*-axis:

Revolving about the *y*-axis:

See Figure 12.4-5.

Revolving about a line *y* = *k*:

Revolving about a line *x* = *h*:

See Figure 12.4-6.

### Example 1

Find the volume of the solid generated by revolving about the *x*-axis the region bounded by the graph of , the *x*-axis, and the line *x* = 5.

Step 1. Draw a sketch. See Figure 12.4-7.

Step 2. Determine the radius of a disc from a cross section.

Step 3. Set up an integral.

Step 4. Evaluate the integral.

Verify your result with a calculator.

### Example 2

Find the volume of the solid generated by revolving about the *x* -axis the region bounded by the graph of , the *x*-axis, and the *y*-axis.

Step 1. Draw a sketch. See Figure 12.4-8.

Step 2. Determine the radius from a cross section.

Step 3. Set up an integral.

Step 4. Evaluate the integral.

Thus the volume of the solid is .

Verify your result with a calculator.

### Example 3

Find the volume of the solid generated by revolving about the *y*-axis the region in the first quadrant bounded by the graph of *y* = *x*^{2}, the *y*-axis, and the line *y* = 6.

Step 1. Draw a sketch. See Figure 12.4-9.

Step 2. Determine the radius from a cross section.

- is the part of the curve involved in the region.

Step 3. Set up an integral.

Step 4. Evaluate the integral.

The volume of the solid is 18.

Verify your result with a calculator.

### Example 4

Using a calculator, find the volume of the solid generated by revolving about the line *y* = 8 the region bounded by the graph of *y* = *x* 2 + 4, the line *y* = 8.

Step 1. Draw a sketch. See Figure 12.4-10.

Step 2. Determine the radius from a cross section.

*r*= 8 -

*y*= 8 - (

*x*

^{2}+ 4) = 4 -

*x*

^{2}

Step 3. Set up an integral.

- To find the intersection points, set

Step 4. Evaluate the integral.

Thus the volume of the solid is .

Verify your result with a calculator.

### Example 5

Using a calculator, find the volume of the solid generated by revolving about the line *y* = –3 the region bounded by the graph of *y* = *e*^{x}, the *y*-axis, the lines *x* = ln 2 and *y* = – 3.

Step 1. Draw a sketch. See Figure 12.4-11.

Step 2. Determine the radius from a cross section.

*r*=

*y*– (–3) =

*y*+ 3 =

*e*

^{ x}+3

Step 3. Set up an integral.

Step 4. Evaluate the integral.

The volume of the solid is approximately 13.7383.

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

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