Volumes of Solids of Revolution

In short: A solid of revolution is made by spinning a curve around an axis, and its volume is found by adding up thin discs. Spun around the x-axis, $V=\pi {\int }_a^b{y}^{2}dx$, and around the y-axis, $V=\pi {\int }_c^d{x}^{2}dy$.

Starting point: what you already know

• Area of a circle: $A=\pi r^2$. A thin coin of radius $r$ and thickness $h$ has volume $\pi r^{2}h$.
• Definite integrals as sums (Mathematics Advanced): ${\int }_a^{b}f(x)dx$ adds up thin strips of height $f(x)$ and width $dx$.
• Integrating powers: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$.
• Making $x$ the subject: for example $y=x^2$ gives $x^2=y$.

Slicing into discs

Take the area under $y=f(x)$ from $x=a$ to $x=b$ and spin it once around the x-axis. Slice the solid at right angles to the axis.

• Each slice is a disc of radius $y$ and thickness $\delta x$.
• Its volume is about $\pi y^2\delta x$.
• Adding the slices and letting $\delta x\to 0$ turns the sum into an integral.

Spinning around the y-axis

Slices are now horizontal, with radius $x$ and thickness $\delta y$. Write $x^2$ in terms of $y$, and use $y$ limits.

Quick summary

• A solid of revolution is cut into thin discs of volume $\pi r^2\delta x$.
• About the x-axis: $V=\pi {\int }_a^b{y}^{2}dx$. About the y-axis: $V=\pi {\int }_c^d{x}^{2}dy$.
• Square first, then integrate. Use limits that match the variable.
• For a hollow solid, subtract the inner squared radius from the outer one.

Quiz