The Big Circle Minus the Small

You'll learnto build a washer as two disc integrals subtracted, to see why the width of a ring is not the radius of a disc, to assign the outer and inner radii by testing a point, and to split where a region stops touching the axis.

When the region does not reach the line it spins about, every slice comes out as a ring. Its area is the big circle minus the small one — not π times the square of the gap between the radii, which for radii 5 and 3 gives 4π against the true 16π. The two agree only when the hole has no size at all, which is exactly the case where the ring was a disc. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · π(R² − r²), AND NEVER π(R − r)² · v2

Subtract the two circle areas, never square the gap between the radii

Before you start

What you're looking at

A region that never reaches the line it is being spun about, and the ring one slice of the solid makes. The outer edge comes from the far boundary and the inner edge from the near one.

The question

How far is each boundary from that line, and how much area is left between the two circles they sweep?

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