Computing volumes for solids of revolution using cylindrical shells（利用柱壳法计算旋转体体积）:
Shell method
柱壳法对于旋转固体体积的计算公式如下：
Setting up the Integral
• Keypoints:
1. When using cylindrical shells, you integrate with respect to the variable that is perpendicular to the axis of rotation.(使用柱壳法时，可以相对于垂直于旋转轴的变量进行积分)
2. The integral can be set up as 2π ∫(a to b) r(x) h(x) dx or2π ∫(c to d) r(y) h(y) dy, depending on the orientation.
例题
Example 1:
Use the shell method to find the volume of the solid generated by revolving the shaded region about the y-axis.
Limit is 0<x<pi
Example 2:
Use the shell method to find the volume of the solid generated by revolving the shaded region about the x-axis.
Example 3:
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.
Example 4:
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. You must include a clearly labeled sketch of the region.
Example 5 :
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis.
Example 6:
use the shell method to find the volume of the solid generated by revolving the region bounded by the give curves about the given lines.
Practice:
Find the volume of the solid generated by the revolving the region about the given axis. Use the shell method. The region bounded by x=3 𝑦, 𝑥 = −3𝑦 𝑎𝑛𝑑 𝑦 = 1 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑦 = 1
📝Summary：
