Volume of a Solid Obtained by Revolving Area About an Axis

Solids of revolution. When a plane region is given one complete rotation ($360^\circ=2\pi$ radians) about a fixed axis lying in its own plane, it sweeps out a solid of revolution. For example, revolving the semicircular region enclosed by $x^2+y^2=a^2$ above the $x$-axis about the $x$-axis generates a sphere of radius $a$; revolving the rectangular region bounded by $y=0,\,y=a,\,x=0,\,x=h$ about the $x$-axis generates a right-circular cylinder of radius $a$ and height $h$.

This section restricts to revolution about the $x$-axis or the $y$-axis. For revolution about the $x$-axis, the revolved plane region lies above the $x$-axis ($y\ge0$); for revolution about the $y$-axis, it lies to the right of the $y$-axis ($x\ge0$).

Derivation (disc method), about the $x$-axis. Let $y=f(x)$, $x$-axis, $x=a,\,x=b$ ($b>a$) bound a region in the first quadrant, with every vertical line between $x=a$ and $x=b$ meeting the curve exactly once. Divide $[a,b]$ into $n$ segments $x_0=a<x_1<\cdots<x_n=b$, $\Delta x=\frac{b-a}n$. On each subinterval, the thin rectangle of height $y_i=f(x_i)$ and width $\Delta x$, revolved about the $x$-axis, sweeps out an elementary cylindrical disc of radius $y_i$ and height $\Delta x$, hence volume $\pi y_i^2\Delta x$ (using "volume of a cylinder $=\pi r^2h$"). Summing all the discs, $\sum \pi y_i^2\,\Delta x$, and letting $n\to\infty,\ \Delta x\to0$, this tends to the volume of the whole solid:

By the identical argument with $x$ and $y$ interchanged, for a curve $x=f(y)$, $y$-axis, and $y=c,\,y=d$ revolved about the $y$-axis,

Standard solids re-derived by these formulas (Examples 9.62-9.69), all worth having on hand as checks:

• Sphere of radius $a$: revolve $y=\sqrt{a^2-x^2}$, $-a\le x\le a$, about the $x$-axis: $V=\pi\int_{-a}^a(a^2-x^2)\,dx=2\pi\int_0^a(a^2-x^2)\,dx=\dfrac43\pi a^3$.
• Right circular cone, base radius $r$, height $h$: revolve the triangular region under $y=\dfrac rh x$, $0\le x\le h$, about the $x$-axis: $V=\pi\int_0^h\left(\dfrac rhx\right)^2dx=\dfrac13\pi r^2h$.
• Spherical cap of height $h$ cut from a sphere of radius $r$: revolve $y=\sqrt{r^2-x^2}$, $r-h\le x\le r$, about the $x$-axis: $V=\pi h^2\!\left(r-\dfrac h3\right)$; in terms of the cap's own base radius $\rho$ (where $\rho^2+(r-h)^2=r^2$), $V=\dfrac{\pi h}{6}\left(3\rho^2+h^2\right)$.
• Ellipsoid, from the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b$) revolved about the major axis ($x$-axis): $V=\dfrac43\pi ab^2$; revolved about the minor axis ($y$-axis) instead: $V=\dfrac43\pi a^2b$. …