Volumes by Cross-Sections, Disks, and Washers

Definition Let $B$ be a plane region with area $A$. A (right) cylinder with base $B$ and height $h > 0$ is the solid swept out when $B$ moves a distance $h$ perpendicular to its own plane. Its volume is

A prism is a cylinder whose base is a polygon. A circular cylinder of radius $r$ has $V = \pi r^2 h$.

Definition Let $S$ be a solid lying between the planes $x = a$ and $x = b$. For each $x \in [a, b]$, the cross section $S_x$ is the intersection of $S$ with the plane perpendicular to the $x$-axis at $x$, and $A(x)$ is its area. The same idea applies with any axis in place of the $x$-axis.

Definition Let $S$ lie between $x = a$ and $x = b$, and suppose its cross-sectional area $A(x)$ is continuous on $[a, b]$. Partition $[a, b]$ into subintervals of widths $\Delta x_i$ and pick $x_i^*$ in each. The volume of $S$ is

Definition If two solids lie between the same two parallel planes and every plane parallel to them cuts both solids in cross sections of equal area, then the solids have equal volume. When the common area function is continuous, this follows at once from the slice formula above. In particular an oblique cylinder, whose base slides along a slanted line instead of straight up, has volume $V = A h$, where $h$ is the perpendicular distance between the base planes.

Definition A pyramid with base area $A$ and height $h$ (perpendicular distance from apex to base plane) has volume

The cross section at distance $y$ from the apex, parallel to the base, is similar to the base with ratio $y/h$, so its area is $A y^2 / h^2$, and

Definition A circular cone with base radius $r$ and height $h$ has volume

It is the solid obtained by rotating the triangle under the line $y = rx/h$ for $0 \le x \le h$ about the $x$-axis. Its cross sections are disks of radius $rx/h$, so

Definition A solid of revolution is the solid swept out when a plane region rotates one full turn about a line in its plane, called the axis of rotation, where the region lies entirely on one side of the axis (it may touch the axis along its boundary). When the region lies between two graphs over an interval of the axis, each cross section perpendicular to the axis is a disk or a washer.

Definition Let $f$ be continuous and nonnegative on $[a, b]$. Rotating the region under $y = f(x)$ about the $x$-axis gives a solid with disk cross sections, so

If $0 \le g(x) \le f(x)$ are continuous on $[a, b]$, rotating the region between them about the $x$-axis gives washers with outer radius $f(x)$ and inner radius $g(x)$, so

Definition A ball of radius $r > 0$ has volume

It is the solid of revolution formed by rotating the half disk under $y = \sqrt{r^2 - x^2}$, $-r \le x \le r$, about the $x$-axis, so

Definition For rotation about the horizontal line $y = k$, the radius of a point on the curve $y = f(x)$ is its distance $\lvert f(x) - k \rvert$ to the axis. For a region $g(x) \le y \le f(x)$ lying on one side of the axis, the outer radius $R(x)$ is the larger of the distances from the axis to the two boundary curves and the inner radius $r(x)$ is the smaller, so

where $[a, b]$ is the interval of $x$ values covered by the region. For a vertical axis, use horizontal slices and integrate in $y$.