Topic 8.7 Volumes with Cross Sections: Squares and Rectangles: integrate the cross-sectional area to find volume when cross sections are squares or rectangles.

What this topic is asking

The College Board (Topic 8.7) finds the volume of a solid whose base is a region in the plane and whose cross sections perpendicular to an axis are squares or rectangles. You integrate the cross-sectional area along the axis.

The volume-by-cross-section principle

A worked square-cross-section volume

Getting the side length right

The crux is writing the side of the cross section from the base region. When cross sections are perpendicular to the $x$-axis, the side is the region's vertical extent at that $x$ (top curve minus bottom curve). When perpendicular to the $y$-axis, the side is the region's horizontal extent at that $y$ (right curve minus left curve), which forces you to write $x$ in terms of $y$ and integrate in $y$. Choosing the wrong variable, or measuring the wrong extent, gives a wrong $A$. Read the problem for which axis the cross sections are perpendicular to, and measure the base region across that direction.

Why squaring the right length matters

For square cross sections, $A(x) = (\text{side})^2$, so any error in the side is amplified by squaring. A common mistake is to use the curve value (like $\sqrt{x}$) as the side when the side is actually the full height between two curves, or to forget the factor of $2$ for a symmetric region spanning both sides of an axis (width $2\sqrt{y}$, not $\sqrt{y}$). The reliable habit is to draw a representative cross section, label its side as a difference of the bounding curves, then square. With the correct side, the integral of $A$ over the base interval gives the volume directly, in cubic units.

Distinguishing from solids of revolution

Volume-by-cross-section solids are not solids of revolution: nothing is spun about an axis. Instead a flat base region is given, and known shapes (here squares or rectangles) are erected on it perpendicular to an axis. The general principle $V = \int A\,dx$ still applies, and the disc and washer methods of the later topics turn out to be the special case where the cross section is a circle or annulus generated by revolution. Recognizing that this topic uses a given cross-section shape, while revolution problems generate circular cross sections, keeps the two families clearly separated, since they read the side or radius from the geometry differently. A telltale sign is the phrase "cross sections perpendicular to the axis are squares", which signals the given-shape method rather than revolution.

Setting up the rectangle case

For rectangular cross sections one dimension is the base region's extent and the other is supplied by the problem, often as a fixed multiple of the first ("the height of each rectangle is twice its base") or a constant. So if the base region's height at $x$ is $h(x)$ and the rectangle's other side is, say, $3$, then $A(x) = 3\,h(x)$; if the other side is twice the base, $A(x) = 2[h(x)]^2$. The key is to read the relationship between the two rectangle dimensions from the wording and express both in terms of the integration variable before integrating. Once $A(x)$ is written correctly, the volume is again just $\int_a^b A(x)\,dx$, the same template as the square case with a different area formula.