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.

8 STEPS · 6 QUICK CHECKS · SUBTRACT THE AREAS, WHICH MEANS THE SQUARES · v1

solve for the crossings · test inside · then subtract the squares

Before you start

What you're looking at

One plane at equal scale, with the pole at the left of centre. Two curves share it: the circle r = 3cos θ, which is a circle of radius 1.5 centred at (1.5, 0), and the cardioid r = 1 + cos θ. The shaded region is inside the circle and outside the cardioid.

The question

The integrand is a difference of squares, and it is the part that is inherited. What is new is the interval: it runs between the angles where the radii are equal, and outside those angles the two curves exchange roles entirely.

Watch for

Both curves pass through the pole and neither meets the other there at a shared angle — the circle arrives at θ = π/2 and the cardioid at θ = π. Setting the radii equal cannot find that intersection, which is why step 1 draws it separately.

Step 1 / 8