A student wants to reduce the size of a photograph that is 12 inches wide. She selects 75% on the photocopy machine. What will b
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I assume the cone has equation (i.e. the upper half of the infinite cone given by
). Take
The volume of the described region (call it ) is
The limits on and
should be obvious. The lower limit on
is obtained by first determining the intersection of the cone and sphere lies in the cylinder
. The distance between the central axis of the cone and this intersection is 1. The sphere has radius
. Then
satisfies
(I've added a picture to better demonstrate this)
Computing the integral is trivial. We have
