5.4 · Volumes with known cross sections

The base width is not the cross-sectional area

For a solid with square sections perpendicular to the x-axis, the side length is the width of the base region at $\displaystyle x$. If the upper boundary is $\displaystyle f$ and lower boundary is $\displaystyle g$, the width is $\displaystyle w(x)=f(x)-g(x)$ and area is $\displaystyle w(x)^2$.

The volume is the integral of the cross-sectional area, not the integral of the base width. The solid does not need to be generated by rotation.

$$V=\int_a^b A(x)\,dx$$

The section shape determines the coefficient

A square of side $\displaystyle w$ has area $\displaystyle w^2$. A semicircle with diameter $\displaystyle w$ has radius $\displaystyle w/2$ and area $\displaystyle \pi w^2/8$. An equilateral triangle of side $\displaystyle w$ has area $\displaystyle \sqrt3\,w^2/4$.

If triangles have a specified height rather than an equilateral shape, use $\displaystyle A=bh/2$. Read whether the base width represents a radius, diameter, side, or another dimension.

Match section orientation to the variable

Sections perpendicular to the x-axis produce $\displaystyle dx$ slices; sections perpendicular to the y-axis produce $\displaystyle dy$ slices. Rewrite the base boundaries in terms of the slice variable.

Check units and endpoint behavior. If a base region narrows to width zero, the cross-sectional area should also approach zero for these similar shapes.