Question

Find the volume of the solid generated by revolving the curve r=a(1+cos θ) about the x-axis

Answer 1

Find the volume of the solid generated by revolving the curve r=a(1+cos θ) about the x-axis

Solution:

The volume of the solid is generated by revolving the upper 1/2 of the centroid of initial line that is theta =0,

the portion that is above the initial line is theta and theta is equal to "0 to pi "

V=∫2/3 pi *r^2*sin a da

a= theta

upper limit a

and lower limit 0

=2pi/3∫a^3*(1-cos a)^3 sina da

now,

put t=1-cos a

sin a da=dt

a=t=0,

a=pi,

t=2

volume=8/3 *pi*a^3