Problem

Consider the region between the circles x² + y² = a² and x² + y² = b² in the first quadrant. Divide this region into two pieces with the curve defined by

in the first quadrant. Find the ratio of the two regions created and sketch them.

Solution

The two circles have radii a and b, and the new curve is an ellipse touching each of the circles. We assume that a < b, but the same arguments hold for a > b.

Let's try to find the area of R1 by integrating with respect to x.

The last integral is one we should know: it is the area of a quarter circle of radius a. Therefore, the area of R1 is

Now the area of R2. This time we integrate with respect to y.

The last integral is the area of a quarter circle of radius b. The area of R2 is therefore

The ratio R2/R1 equals

which simplifies to