[7.2] Find the exact volume of the solid obtained by revolving the
region R bounded by the
curve y=3square root 3
and the lines with equations x=1 x=8
and y=0 about the line with equation:
I y=0
ii y=3
Solution:
a) when region is rotated about y=0,
The volume of solid can be obtained from the below integral,
V=integration(with limits x=1 to 8)pi y^2 dx
V=integration(with limits x=1 to 8) 9 pi x dx
V=9 pi (15/2)=135pi/2
b) When region is rotated about y=3,
The volume of solid can be obtained from the below integral,
V=integration(with limits x=1 to 8)pi (y-3)^2 dx
V=integration(with limits x=1 to 8) pi (y^2+9-6y) dx
V=integration(with limits x=1 to 8) 9 pi (9x+9-18sqrt(x)) dx
V=(717/2-(3*2^(13/2)))pi
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