3.6 Integration as the Summation of Volumes

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

$V_x=\pi {\displaystyle {\int }_a^b{y}^{2}dx}$

(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

$$V_y=\pi {\displaystyle {\int }_a^b{x}^{2}dy}$$