How do I determine the volume of the solid obtained by revolving the curve r=3sin(θ) around the polar axis?

1 Answer
Wataru
Nov 12, 2014

Let us look at the polar curve r=3sinθ.

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The above is actually equivalent to the circle with radius 32, centered at (0,32), whose equation is:

x2+(y−32)2=(32)2

by solving for y, we have

y=±√(32)2−x2+32

By Washer Method, the volume of the solid of revolution can be found by

V=π∫32−32⎡⎢⎣⎛⎝√(32)2−x2+32⎞⎠2−⎛⎝−√(32)2−x2+32⎞⎠2⎤⎥⎦dx

by simplifying the integrand,

=6π∫32−32√(32)2−x2dx

since the integral can be interpreted as the area of semicircle with radus 32,

=6π⋅π(32)22=27π24

I hope that this was helpful.

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