Determining the Volume of a Solid of Revolution

Key Questions

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

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

    enter image source here

    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.

    Wataru · 1 · Nov 12 2014

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Polar Curves