Volumes of Solids of Revolution

Questions involving the area of a region between curves, and the volume of a solid formed when this region is rotated about a horizontal or vertical line, appear regularly on both the AP Calculus AB and BC exams. Students have difficulty when the solid is formed by use a line of rotation other than the x- or y-axis.

These types of volume are part of the type of volume problems students must solve on the AP test. Students should find the volume of a solid with a known cross section. The Shell method is not part of the AB or the BC course of study anymore.

The four examples in the Curriculum Module use the disk method or the washer method.

Example 1 Line of Rotation Below the Region to be Rotated Picture the solid (with a hole) generated when the region bounded by and are revolved about the line y = -2. First find the described region Then create the reflection over the line y=-2

Example 1 Think about each of the lines spinning and creating the solid. Draw one representative disk. Draw in the radius.

Example 1 Find the radius of the larger circle, its area and the volume of the disk.

Sum up these cylinders to find the total volume The larger the number of disks and the thinner each disk, the smoother the stack of disks will be. To obtain a perfectly smooth solid, we let n approach infinity and Δx approach 0.

The points of intersection can be found using the calculator. Store these in the graphing calculator (A=-1.980974,B=0.13793483) (C=0.44754216,D=1.5644623) Write an integral to find the volume of the solid.

Example 1 Find the radius of the smaller circle, its area and the volume of the disk.

Sum up these cylinders to find the total volume The larger the number of disks and the thinner each disk, the smoother the stack of disks will be. To obtain a perfectly smooth solid, we let n approach infinity and Δx approach 0.

Using the points of intersection write a second integral for the inside volume. (A=-1.980974,B=0.13793483) (C=0.44754216,D=1.5644623)

Example 1 The final volume will be the difference between the two volumes.

Example 2 Line of Rotation Above the Region to be Rotated Rotate the same region about y = 2 Notice that

Example 2 Line of Rotation Above the Region to be Rotated The area of the larger circle is

Example 2 Line of Rotation Above the Region to be Rotated The sum of the volumes is

Example 2 Line of Rotation Above the Region to be Rotated The area of the smaller circle is

Example 2 Line of Rotation Above the Region to be Rotated The sum of the volumes is

Example 2 Line of Rotation Above the Region to be Rotated The volume of the solid is the difference between the two volumes

Example 3 Line of Rotation to the Left of the Region to be Rotated Line of Rotation: x = -3 Use the same two functions Create the reflection Draw the two disks and mark the radius

Example 3 Line of Rotation to the Left of the Region to be Rotated The radius will be an x-distance so we will have to write the radius as a function of y.

Example 3 Line of Rotation to the Left of the Region to be Rotated The radius of the larger disk is 3 + the distance from the y-axis or 3 + (ln y) Area of the larger circle is

Example 3 Line of Rotation to the Left of the Region to be Rotated Volume of each disk:

Example 3 Line of Rotation to the Left of the Region to be Rotated The radius of the smaller disk is 3+ the distance from the y-axis or 3 + (y2 – 2) Area of the larger circle is

Example 3 Line of Rotation to the Left of the Region to be Rotated Volume of each disk:

Example 3 Line of Rotation to the Left of the Region to be Rotated Difference in the volume is

Example 4 Line of Rotation to the Right of the Region to Be Rotated Line of Rotation: x = 1 Create the region, reflect the region and draw the disks and the radius

Example 4 Line of Rotation to the Right of the Region to Be Rotated Notice the larger radius is 1 + the distance from the y-axis to the outside curve. The distance is from the y-axis is negative so the radius is

Example 4 Line of Rotation to the Right of the Region to Be Rotated Area of Larger disk: The volume of the disk is

Example 4 Line of Rotation to the Right of the Region to Be Rotated Volume of all the disks are

Example 4 Line of Rotation to the Right of the Region to Be Rotated Area of smaller disk: The volume of the disk is

Example 4 Line of Rotation to the Right of the Region to Be Rotated Volume of all the disks are

Example 4 Line of Rotation to the Right of the Region to Be Rotated Find the difference in the volumes