Cross-Section
A cross-section is the 2-D shape obtained by cutting through a 3-D solid with a plane.
Definition
A cross-section is the 2-D shape you see when you slice straight through a 3-D object, like cutting a loaf of bread and looking at the slice face; formally, it is the intersection of a solid $S$ with a plane $H$, $S \cap H$. For a prism or cylinder, horizontal cross-sections are congruent to the base, while oblique cuts produce different shapes, and cross-sections are the basis of Cavalieri's principle: if every cross-section of two solids at the same height has equal area, the solids have equal volume. For a solid of revolution generated by $f(x) > 0$ rotated about the x-axis, the cross-section at position $x$ is a disk of radius $f(x)$ and area $\pi f(x)^2$, giving the disk method for volume, $V = \int \pi f(x)^2 \, dx$ (the washer method handles solids with holes).
Example
Slice a cylinder horizontally and you see a circle; slice it vertically through the center and you see a rectangle. A cube cut by a diagonal plane through $4$ vertices produces a rectangular cross-section, while cutting through $6$ edge midpoints at an angle gives a regular hexagon, and a sphere cut by any plane gives a circle (a great circle if through the center). For a solid of revolution from $y = \sqrt{x}$ on $[0,4]$ rotated about the x-axis, each cross-section at $x$ is a disk of radius $\sqrt{x}$ and area $\pi x$, so $V = \int_0^4 \pi x \, dx = \pi\left[\frac{x^2}{2}\right]_0^4 = 8\pi$.
Key Insight
The shape of a cross-section depends on the angle of the cut: slicing a cone at different angles creates the four conic sections, circle, ellipse, parabola, and hyperbola, one of the most famous ideas in all of geometry. Archimedes used Cavalieri's principle to find the volume of a sphere, $2000$ years before calculus, by comparing its cross-sections to those of known solids; the disk and shell methods for volumes of revolution are two equivalent ways to decompose a solid into thin cross-sections or shells, and the choice between them is a recurring theme in applied calculus.