A Difference of Squares, in Polar Form

You'll learnto build the region between two polar curves out of the gap between two sectors, to subtract the squares of the radii rather than the radii themselves, to find the interval by solving for the intersection angles and to check the pole separately, to decide which curve is outer with a test angle, and to split the interval wherever that assignment stops being true.

One half the integral of R squared minus r squared, and never one half the integral of the quantity R minus r, squared. This is Topic 8.11's washer in polar clothing, and it fails for the same reason: a hole is an area to subtract, not a length to shorten a radius by. The genuinely new work is finding the interval, which runs between the angles where the two curves cross. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · SUBTRACT THE AREAS, WHICH MEANS THE SQUARES · v2

subtract the areas · which means subtracting the squares

Before you start

What you're looking at

One plane at equal scale with the pole at the left of center. A pink circle and an indigo cardioid share that pole, and the shaded band between them is the region.

The question

The region is the gap between two curves. Where does it start and stop, and what exactly gets subtracted from what?

Step 1 / 12