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## Problem

The bicylinder is a solid formed by the intersection of two cylinders of equal cross-sections. Use integration to determine the volume of the bicylinder if the cross-sections of the intersecting cylinders are circles of radius .

## Solution

The first thing to consider are the level-curves in the direction of integration. Let the integration be carried out in the y-axis. The cross-sections in the y-axis are squares that vary under the constraint of a circle with radius , which is the circle . The area of the cross-sectional squares is determined by the formula

.

Thus the cross-sectional areas are

.

To determine the volume, integrate in the y-axis from one end of the circle to the other end.