8.7 Volumes with Cross Sections: Squares and Rectangles

Slicing solids into cross sections

For solids with known cross sections perpendicular to an axis, we find volume by integrating the area of each cross section along that axis. The side length of each square/rectangle cross section usually comes from the distance between two curves, like top minus bottom or right minus left.

Square cross sections

If cross sections perpendicular to the x-axis are squares with side length s(x) = f(x) − g(x), the area is s(x)², so Volume = ∫[a,b] (f(x) − g(x))² dx. Always find the side length first, then square it before integrating.

Rectangular cross sections

For rectangles, you need both a base and a height, often given as a ratio or separate function (e.g., height = 2×base). Area = base × height, expressed in terms of x or y, then integrate: Volume = ∫[a,b] base(x)·height(x) dx. Setting up the correct expression for each dimension is key.

Key terms

Volume formula for square cross sections

V = ∫[a,b] s(x)² dx, where s(x) is the side length (distance between the boundary curves) at each x.

Volume formula for rectangular cross sections

V = ∫[a,b] base(x)·height(x) dx, where base and height are given or related functions of x (or y).

Finding side length between two curves

Subtract: top curve minus bottom curve (or right minus left), since length must be nonnegative.

Slicing perpendicular to the y-axis

Express all functions and the differential in terms of y, then integrate with dy instead of dx.

Key setup step before integrating

Determine the cross-sectional area as a function of the axis variable, then integrate that area function over the interval.

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. 8
2. 16
3. 8π
4. 32/3