A Slice That Is a Circle

You'll learnto read a solid of revolution as one more cross-section problem, to square the radius before integrating rather than after, to measure the radius from the axis, and to check whether a solid disc was ever the right shape.

Spin a region about a line and every slice of the solid is a circle, so this is Topic 8.7 with the area of a circle in place of the area of a square. Two things go wrong: where the squaring happens, and whether a solid disc was ever the right shape. The second is the dangerous one, because that integral is well formed, evaluates cleanly, and is wrong. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · SQUARE FIRST, AND ASK IF IT TOUCHES THE AXIS · v2

The radius is measured from the axis, and a disc has to touch it

Before you start

What you're looking at

A region, the line it is being spun around, and the mirror image the spin sweeps out. Beside it, one slice of the solid drawn face on: a circle, or a circle with a hole.

The question

How far is the boundary from the axis here, and does anything stand between the axis and the region?

Step 1 / 12