What Is the Area of One Slice?

You'll learnto build a volume as an integral of one slice's area, to take the slice's side from the distance between two boundaries rather than from one curve, and to keep a square, a scaled rectangle and a fixed-height rectangle apart.

A solid built on a flat base, sliced perpendicular to an axis, is just an integral of one slice's area. Two things have to be right: the side, which is the distance between the two boundaries of the base, and the formula that turns that side into an area. Everything in the next six topics is that sentence with a different shape in it — discs and washers included, where the shape happens to be a circle.

8 STEPS · 6 QUICK CHECKS · THE SIDE IS A DISTANCE, THEN THE SHAPE IS A FORMULA · v1

V = ∫ A(x) dx ⇒ s = y_top − y_bottom ⇒ square s², rectangle k s² or h s

Before you start

What you're looking at

Two pictures of the same solid. On the left is its flat base, seen from above, with one slice's footprint marked. On the right is that slice itself, seen face on, at its own scale.

The question

How long is the segment the base cuts out at this position, and what area does the named shape have when that segment is its base?

Watch for

The side is the gap between two boundaries. It equals the value of one curve only when the other boundary is the axis, which is true in most first examples and false in general.

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