The Limits Are Where the Curve Finishes

You'll learnto build a polar area from the sector that a small change in angle sweeps, to keep the one-half and the square where they belong, to find a petal's limits by solving for where the curve returns to the pole, to recognise when a full revolution of the angle draws a region more than once, and to check any polar area against a circle whose area you already know.

The formula is one half the integral of r squared, and it is the easy half. The hard half is the limits, because a polar curve can trace the same region twice without anything in its equation warning you. In every earlier topic the interval was where the region is. Here it is where the curve finishes drawing it once, and those are not the same interval.

8 STEPS · 6 QUICK CHECKS · A FULL TURN OF THE ANGLE IS OFTEN TOO MUCH · v1

solve for the pole · then integrate between two consecutive answers

Before you start

What you're looking at

One plane at equal scale, origin at the centre. The running curve is the rose r = 2sin 3θ. Its three petals point at 30°, 270° and 150°, and the middle one is drawn where r is negative — which is why it appears opposite the direction its angles name.

The question

The integrand is never in doubt. What is in doubt is the interval, and the same integrand over three different intervals gives three different answers here: one petal, three petals, and three petals counted twice.

Watch for

In step 4 the sweep does not stop at half a revolution. Watch the shaded region get darker rather than bigger, and watch the number keep climbing while the picture stops changing shape. That is the whole topic in one beat.

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