【高数定积分求解旋转体体积】 —— （上）高等数学|定积分|柱壳法|学习技巧

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 or 2π ∫(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：