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 recognize 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. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · A FULL TURN OF THE ANGLE IS OFTEN TOO MUCH · v2

the formula is the easy half · the limits are the work

Before you start

What you're looking at

One plane at equal scale, with the pole at its center. A turning radius draws a three-petalled rose, and the shaded fan behind it is the area swept so far.

The question

The area formula never changes. So where do the two limits come from, and what happens if you turn the angle further than the curve needs?

Step 1 / 12