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.

Revolving a region about an axis makes every cross section a circle, so this is Topic 8.7 with A equal to π r squared. The radius is the distance from the axis to the curve, and the two things that go wrong are where π and the square live, and whether a solid disc was ever the right shape at all. The second is the dangerous one: the integral is well formed, evaluates cleanly, and is wrong.

8 STEPS · 6 QUICK CHECKS · SQUARE THE RADIUS FIRST, AND ASK IF IT TOUCHES THE AXIS · v1

V = π ∫ r² dx ⇒ r is measured from the axis ⇒ a disc needs the region 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 resulting solid, drawn face on: a circle, or a circle with a hole.

The question

How far is the boundary from the axis at this position, and does anything stand between the axis and the region there?

Watch for

The axis of revolution is drawn as its own object. Here it happens to be one of the coordinate axes; from Topic 8.10 it will not be, and the radius is a distance from it either way.

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