# 3.6 Integration as the Summation of Volumes

3.6 Integration as the Summation of Volumes

(1).

The volume of the solid generated when the region enclosed by the curve y = f(x), the x-axis, the line x = a and the line x = b is revolved through 360° about the x-axis is given by

${V}_{x}=\pi {\int }_{a}^{b}{y}^{2}dx$

(2).

The volume of the solid generated when the region enclosed by the curve x = f(y), the y-axis, the line y = a and the line y = b is revolved through 360° about the y-axis is given by
${V}_{y}=\pi {\int }_{a}^{b}{x}^{2}dy$

Example 1:
Find the volume generated for the following diagram when the shaded region is revolved through 360° about the x-axis.

Solution:
Volume generated, Vx

Example 2:
Find the volume generated for the following diagram when the shaded region is revolved through 360° about the y-axis.

Solution:
Volume generated, Vy