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 =π 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 =π 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
V x =π a b y 2 dx V x =π 2 4 ( 3x 8 x ) 2 dx V x =π 2 4 ( 3x 8 x )( 3x 8 x )dx V x =π 2 4 ( 9 x 2 48+ 64 x 2 )dx V x =π [ 9 x 3 3 48x+ 64 x 1 1 ] 2 4 V x =π [ 3 x 3 48x 64 x ] 2 4 V x =π[ ( 3 (4) 3 48(4) 64 4 )( 3 (2) 3 48(2) 64 2 ) ] V x =π( 16+104 ) V x =88π uni t 3


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
V y =π a b x 2 dy V y =π 1 2 ( 2 y ) 2 dy V y =π 1 2 ( 4 y 2 )dy V y =π 1 2 4 y 2 dy V y =π [ 4 y 1 1 ] 1 2 =π [ 4 y ] 1 2 V y =π[ ( 4 2 )( 4 1 ) ] V y =2π uni t 3