Solids With Rectangular Cross Sections

From a Base Region to Rectangular Slices

A solid can be described by the shapes of its cross sections, even when its full three-dimensional boundary is difficult to write down. For solids with rectangular cross sections, each slice is a rectangle. One side of that rectangle lies along a segment of a specified base region; the other side is determined by a height rule.

The key is to identify these two side lengths separately. The segment across the base is not automatically the height of the rectangle, and the rectangle is not necessarily a square. Once both lengths are expressed at each slice position, their product gives the cross-sectional area. Integrating those areas gives the volume.

In a common setup, the base region is between \(y=f(x)\) and \(y=g(x)\), with \(f(x)\leq g(x)\) throughout the interval. A vertical slice has length \(s(x)=g(x)-f(x)\). The height rule then supplies the other side. For example, the rule might say that the height is a fixed constant, a specified multiple of the base segment, or a function of \(x\).

Building the Area Function

Before integrating, translate the geometry into a formula. Keep the base length and the height rule visible as separate factors until the area has been formed. This makes it easier to catch a wrong boundary order or a mistaken assumption that the rectangles are squares.

The height must be a nonnegative length throughout the interval, as must the base length. If either is zero at an endpoint, the cross section there has zero area; this is consistent with the volume integral. If a proposed height formula becomes negative, it does not describe a physical height on that part of the interval and the model must be reconsidered.

Worked Example: Height Twice the Base Length

Why the Integral Gives the Volume

The formula follows by approximating the solid with thin slabs. If the interval \([a,b]\) is divided into small subintervals, a slab of thickness \(\Delta x_i\) near \(x=c_i\) has approximately the cross-sectional area \(A(c_i)\). Its approximate volume is \(A(c_i)\Delta x_i\). Adding the slab volumes gives a Riemann sum.

This reasoning explains why the height rule belongs inside the area function. The integral accumulates the areas of the cross sections, not just the widths of the base slices. A constant height can be factored out of the integral, but a height that varies with \(x\) must remain in the integrand.

Worked Example: Constant Rectangle Height

Worked Example: A Position-Dependent Height

A Common Mistake: Assuming the Rectangles Are Squares

A square is a special kind of rectangle, not the default cross section. In the first example the height was explicitly twice the base length. If we incorrectly use the square area \(s(x)^2\), we would calculate

That is not the stated solid: it silently replaces the height \(2(2-x)\) with \(2-x\). The correct area is \(s(x)\cdot 2s(x)=2(2-x)^2\), so the correct volume is twice the incorrect value, \(\frac{16}{3}\text{ m}^3\). The correction is to read the height rule before writing the area.

Checks Before Finishing a Model

A reliable volume model keeps the geometry and the calculus connected. Before evaluating the integral, check the following:

• The interval covers the entire base region for the chosen slices.
• The base length is upper boundary minus lower boundary, and is nonnegative throughout.
• The other rectangle side follows the stated height rule; it is not assumed equal to the base length.
• The area is the product of the two side lengths, and the units are squared.
• The definite integral accumulates area over the slice direction, producing cubic units.
• The integrand and resulting volume are nonnegative for a bounded physical solid.

If the region or height rule is described in separate pieces, write the corresponding area formula on each piece and integrate each part over its own interval. The key modeling step remains the same: determine both sides of each rectangle at a slice position, then multiply them.