Determining the volume of solids of revolution is an important topic in integral calculus. Not all solids are spherical in nature, so calculating the volumes of these solids is complicated. There are two basic methods of calculating the volume of a solid - the disk method and the shell method. We will discuss the disk method and a variation of it called the washer method.

When you rotate a curve about the x-axis, you generate a solid. If you take a vertical slice of the solid, the slice is circular. Since the slice is circular, then it has a radius and the radius is the distance from the x-axis to the curve. (See figure 1) Also, since it is circular, then the area of the slice can be calculated by using the formula for the area of a circle. Recall that the formula for the area of a circle is A =p r ². So the area of a slice will be A = p [R (x)] ².

figure 1

If I slice up the solid into an infinite number of slices, then sum up all the areas from x = a to x = b the result will be the volume of the solid. Well, this is the tedious method, so from the concept of a Riemann sum, we get the following formula for calculating the volume of a solid. The formula for finding the volume of solids of revolution about the x-axis is the following.

There are three steps in calculating the volume of the solid, and they are the following.

1.

Draw the region and identify the radius function R (x).

2.

Square R (x) and multiply by p

3.

Integrate to find the volume.

EXAMPLE 1:

Find the volume of the solid generated by revolving the region bounded by the curves y = x - x ² and y = 0 about the x-axis.

STEP 1: Draw the region and identify the radius function R (x).

The radius function for this solid is R (x) = x - x ². Now we need to find the bounds of integration. To do this, we will set y = x - x ² and y = 0 equal to each other and solve for x.

x - x ² = 0 ® x(1 - x) = 0 ® x = 0 or x = 1

STEP 2: Square R (x) and multiply by p .

STEP 3: Integrate to find the volume.

 EXAMPLE 2:

Find the volume of the solid generated by revolving the region bounded by the curves y = 0, x = 0, x = 2, and y = x ⁴ - 4x ² + 8 about the x-axis.

STEP 1: Draw the region and determine the radius function R (x).

The radius for this solid is R (x) = x ⁴ - 4x ² + 8.

Now let us determine the bounds of integration. In this problem they are given to us. The bounds of integration are x = 0 and x = 2.

STEP 2: Square R (x) and multiply by p .

STEP 3: Integrate to find the volume.

Since we are rotating the curve about the y-axis, the radius, R (y) is the distance from the y-axis to the curve. (see figure 2) The horizontal slices are still circular, and the area of a slice is p [R (y)] ².

The volume of the solid generated about the y-axis is the following formula.

The steps for solving this type of problem are the same finding the volume of the solid generated about the x-axis.

figure 2

EXAMPLE 3:

Find the volume of the solid generated by revolving the region bounded by the curves

about the y-axis.

STEP 1: Draw the region and identify the radius function R (y).

The bounds of integration are given to us. They are y = 0 and y = p / 2.

STEP 2: Square R (y) and multiply by p .

STEP 3: Integrate to find the volume.

EXAMPLE 4:

Find the volume of the solid generated by revolving the region bounded by the curves

about the y-axis.

STEP 1: Draw the region and identify the radius function R (y).

The bounds of integration are given to us in the problem. They are y = 0 and y = 3.

STEP 2: Square R (y) and multiply by p .

STEP 3: Integrate to find the volume.

The washer method is an adaptation of the disk method for finding the volume of a solid of revolution. It is used when the solid does not border on or cross the axis of revolution. The cross sections look like washers instead of disks. When using the washer method, you must determine what are the inner and the outer radii. The outer radius will be the radius that is farther from the axis of rotation. The formula for calculating this volume is the following.

The steps needed to complete this type of problem are the following.

1.

Draw the region and determine the outer and inner radii.

2.

Find the limits of integration.

3.

Integrate.

EXAMPLE 5:

Find the volume of the solid generated by revolving the curves y = x ² + 1 and y = x + 3 about the x-axis.

STEP 1: Draw the region and determine the outer and inner radii.

The outer radius, R (x) is the function y = x + 3, and the inner radius, r (x) is the function y = x ² + 1.

STEP 2: Find the limits of integration.

To find the limits of integration, we must solve for the intersection points of the two graphs. To do this, set the two equations equal to each other and solve for x.

x +3 = x ² + 1 ® x ² - x - 2 = 0 ® (x - 2)(x + 1) = 0 ® x = 2 and x = -1

STEP 3: Integrate.

EXAMPLE 6:

Find the volume of the solid generated by revolving the region enclosed by the triangle with vertices (1, 0), (2, 1), and (1, 1) about the y-axis.

STEP 1: Draw the region and determine the outer and inner radii.

The outer radius, R (y) is the function x = y + 1, and the inner radius, r (y) is the function x = 1. You are probably wondering how I came up with the equation x = y + 1, well here are the steps I took to find it.

1. I found the slope of the line. The slope of this line is m = 1.
2. Used the point - slope formula for the equation of a line. Then I solved for x in terms of y.

STEP 2: Find the limits of integration.

The limits of integration can be easily determined by looking at the graph. The limits of integration are y = 0 and y = 1.

STEP 3: Integrate.