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 = and the line x = b is revolved through 360° about the x-axis is given by

V x = π a b y 2 d x


(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 d y


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 d x V x = π 2 4 ( 3 x 8 x ) 2 d x V x = π 2 4 ( 3 x 8 x ) ( 3 x 8 x ) d x V x = π 2 4 ( 9 x 2 48 + 64 x 2 ) d x V x = π [ 9 x 3 3 48 x + 64 x 1 1 ] 2 4 V x = π [ 3 x 3 48 x 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 = 88 π u n i 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 d y V y = π 1 2 ( 2 y ) 2 d y V y = π 1 2 ( 4 y 2 ) d y V y = π 1 2 4 y 2 d y V y = π [ 4 y 1 1 ] 1 2 = π [ 4 y ] 1 2 V y = π [ ( 4 2 ) ( 4 1 ) ] V y = 2 π u n i t 3


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