Rotating Circles

Calculus Level 3

The volume of the 3-dimensional structure formed by rotating the circle x 2 + ( y − 5 ) 2 = 1 x^2 + (y-5)^2 = 1 around the x-axis can be expressed as V = a π 2 V = a\pi^2 . What is the value of a a ?


The answer is 10.

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Calvin Lin by Calvin Lin

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1 solution

A K
May 2, 2014

The shape formed is a torus with minor radius of r r = 1 1 , i.e. the circle radius, and major radius of R R = 5 5 , i.e. the distance from the x-axis and the centre of the circle.

V o l u m e = ( π r 2 ) ( 2 π R ) = 10 π 2 Volume = (\pi r^{2})(2\pi R) = \boxed{10 \pi^{2}}

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