Chapter 6 | Applications of Integration
645
b
V = ∫
⎤ ⎦ 2 dx
π ⎡
⎣ f ( x )
a
4 π [ x ] 2 dx = π ∫ 1 4
= ∫ = π 2
xdx
1
x 2 |
1 4
= 15 π
2 .
The volume is (15 π )/2 units 3 .
6.8 Use the disk method to find the volume of the solid of revolution generated by rotating the region between the graph of f ( x ) = 4− x and the x -axis over the interval [0, 4] around the x -axis.
So far, our examples have all concerned regions revolved around the x -axis, but we can generate a solid of revolution by revolving a plane region around any horizontal or vertical line. In the next example, we look at a solid of revolution that has been generated by revolving a region around the y -axis. The mechanics of the disk method are nearly the same as when the x -axis is the axis of revolution, but we express the function in terms of y and we integrate with respect to y as well. This is summarized in the following rule. Rule: The Disk Method for Solids of Revolution around the y -axis Let g ( y ) be continuous and nonnegative. Define Q as the region bounded on the right by the graph of g ( y ), on the left by the y -axis, below by the line y = c , and above by the line y = d . Then, the volume of the solid of revolution formed by revolving Q around the y -axis is given by (6.4) V = ∫ c d π ⎡ ⎣ g ( y ) ⎤ ⎦ 2 dy .
The next example shows how this rule works in practice.
Example 6.9 Using the Disk Method to Find the Volume of a Solid of Revolution 2
Let R be the region bounded by the graph of g ( y ) = 4− y and the y -axis over the y -axis interval [0, 4]. Use the disk method to find the volume of the solid of revolution generated by rotating R around the y -axis. Solution Figure 6.20 shows the function and a representative disk that can be used to estimate the volume. Notice that since we are revolving the function around the y -axis, the disks are horizontal, rather than vertical.
Made with FlippingBook - professional solution for displaying marketing and sales documents online