Cross-Section Volume Setup Workshop

From a Planar Region to a Solid

A planar region can describe the base of a three-dimensional solid, but the base alone does not determine the solid's volume. We also need to know the shape of each cross-section. If sections perpendicular to a chosen axis are squares, triangles, or semicircles, their areas vary as we move along that axis. Adding those changing areas across the base gives the volume.

The central task is to make the geometry precise before integrating: choose the slice direction, find the interval of slices, and determine the length of the segment cut from the base region by each slice. That segment is the side length, base, or diameter of the specified cross-section.

Why the Area Integral Gives Volume

Divide the interval of slices into short pieces. Over a small interval of width \(\Delta x\), the solid is approximately a thin slab whose cross-sectional area is close to \(A(x_i^*)\) at some sample point \(x_i^*\). Its approximate volume is \(A(x_i^*)\Delta x\). Adding these slab volumes gives an approximation to the whole volume.

As the widths shrink, these sums approach the definite integral. This is the same limit of sums that defines integration, now applied to cross-sectional areas. The reasoning works when the areas vary continuously, and also piece by piece when the solid has finitely many changes in its description and the corresponding integrals are split at those changes.

The cross-sectional shape determines the formula for \(A\); the base region determines the length used in that formula. Keep these roles separate. For instance, the same chord length produces different areas for a square and a semicircle.

Finding the Slice Direction and Chord

A slice perpendicular to the \(x\)-axis is vertical when viewed in the base region. At a fixed \(x\), its intersection with the region is usually a segment running from a lower boundary curve to an upper boundary curve. Its length is upper \(y\)-value minus lower \(y\)-value. A slice perpendicular to the \(y\)-axis is horizontal; its chord length is right \(x\)-value minus left \(x\)-value.

The integration bounds are the endpoints of the region's projection onto the axis along which slices move. Thus vertical slices use the smallest and largest \(x\)-values in the region, while horizontal slices use its smallest and largest \(y\)-values. If one slice direction makes the boundary description or chord calculation awkward, consider the other direction—but use the direction specified by the solid's cross-section planes.

Area Formulas for Three Cross-Sections

Let \(s\) denote a square's side length, \(b\) the base of an equilateral triangle, and \(d\) the diameter of a semicircle. When the chord is specified as that side, base, or diameter, the cross-sectional areas are:

The triangle formula follows from its height. An equilateral triangle of side \(b\) has height \(\frac{\sqrt{3}}{2}b\), so its area is \(\frac{1}{2}b\left(\frac{\sqrt{3}}{2}b\right)=\frac{\sqrt{3}}{4}b^2\). For a semicircle, the radius is half its diameter, so its area is half the area of a circle with radius \(d/2\): \(\frac{1}{2}\pi(d/2)^2=\frac{\pi}{8}d^2\).

A Common Chord-Length Mistake

Suppose a region lies between \(y=x\) and \(y=2x\) for \(0\le x\le1\), with square cross-sections perpendicular to the \(x\)-axis. The vertical chord has length \(2x-x=x\), so its square area is \(x^2\). A common error is to use the upper boundary \(2x\) alone as the side length. That would give the incorrect volume \(\int_0^1(2x)^2\,dx=\frac{4}{3}\).

The boundary curve's height is not generally the chord length. The chord is the distance between the two boundaries. Here the correct volume is \(\int_0^1 x^2\,dx=\frac{1}{3}\). Indeed, an antiderivative is \(x^3/3\), whose derivative is \(x^2\), and evaluating at \(0\) and \(1\) gives \(1/3\). This check also makes the geometric error visible: at each \(x>0\), the actual chord is shorter than \(2x\).

Putting the Setup Together

Before evaluating a cross-section volume integral, make the geometry explicit. A short written record of the slice direction, bounds, and chord length helps prevent errors in both the integral and its interpretation.

This method matters because it separates the geometric model from the integration. If a volume seems implausible, you can check whether the bounds cover the whole base, whether the chord is a difference of boundary values, and whether the correct shape formula was used. A correct integral evaluates the solid described by the setup; a mistaken chord or shape can produce a perfectly computed answer to a different solid.