8.7 Volumes: Square & Rectangle Cross Sections

The cross-section method

Volume by cross sections is V = ∫[a to b] A(x) dx, where A(x) is the cross-sectional area at x. The side length is often the distance between two curves.

Squares & rectangles

For a square cross section with side s(x): A = [s(x)]². For a rectangle: A = base × height.

Key terms

Cross-section volume

V = ∫[a to b] A(x) dx.

A(x)

The cross-sectional area at x.

Square section

A = [s(x)]².

Rectangle section

A = base × height.

Side length

Often the distance between two curves.

Try an AP-style question

The base is the region under y = √x on [0, 4]. Cross sections ⊥ to the x-axis are squares. What is the volume?

1. 16
2. 8
3. 32/3
4. 8π