Volumes of revolution

Volumes of solids formed by rotating a region about the x-axis or the y-axis, the formulas V = π∫y²dx and V = π∫x²dy, the volumes of a cone and a sphere, and the volume between two curves.

The formulas

Rotating a region about an axis sweeps out a solid. Slicing it at right angles to the axis gives thin discs, each of radius $y$ (or $x$) and thickness $dx$ (or $dy$). Adding the discs gives:

$$\text{About the } x\text{-axis: } V = \pi\int_a^b y^2\,dx \qquad \text{About the } y\text{-axis: } V = \pi\int_c^d x^2\,dy$$

The region and its mirror image

Rotating the shaded region under y = √x about the x-axis sweeps out a bowl-shaped solid. The dashed curve shows the lower edge of the solid.

Try this

What is the radius of the disc at $x = a$?

Example 1●●

Find the volume when the region under $y = x^2$ from $x = 0$ to $x = 2$ is rotated about the $x$-axis.

$V = \pi\displaystyle\int_0^2 x^4\,dx = \pi\Big[\tfrac{x^5}{5}\Big]_0^2 = \tfrac{32\pi}{5}$.

Example 2: a sphere●●●

Prove that a sphere of radius $r$ has volume $\tfrac43\pi r^3$.

Rotate the semicircle $y = \sqrt{r^2 - x^2}$, $-r \le x \le r$, about the $x$-axis.

$V = \pi\displaystyle\int_{-r}^{r} (r^2 - x^2)\,dx = 2\pi\Big[r^2 x - \tfrac{x^3}{3}\Big]_0^{r} = 2\pi\big(r^3 - \tfrac{r^3}{3}\big) = \tfrac43\pi r^3$.

Example 3: about the y-axis●●

The region between $y = x^2$, the $y$-axis and $y = 4$ is rotated about the $y$-axis. Find the volume.

$x^2 = y$, so $V = \pi\displaystyle\int_0^4 y\,dy = \pi\Big[\tfrac{y^2}{2}\Big]_0^4 = 8\pi$.

Between two curves

If the region lies between an outer curve $y_1$ and an inner curve $y_2$, each slice is a washer: $V = \pi\displaystyle\int_a^b \big(y_1^2 - y_2^2\big)\,dx$.

Example 4●●●

The region between $y = x$ and $y = x^2$ is rotated about the $x$-axis. Find the volume.

The curves meet at $x = 0$ and $x = 1$, and $x \ge x^2$ between them.

$V = \pi\displaystyle\int_0^1 (x^2 - x^4)\,dx = \pi\big(\tfrac13 - \tfrac15\big) = \tfrac{2\pi}{15}$.

Flashcards

Volume about the $x$-axis

$\pi\displaystyle\int_a^b y^2\,dx$

Volume about the $y$-axis

$\pi\displaystyle\int_c^d x^2\,dy$

Volume between two curves

$\pi\displaystyle\int_a^b \big(y_1^2 - y_2^2\big)\,dx$

Practice

Find the volume when the region under $y = x^2$ from $0$ to $2$ is rotated about the $x$-axis. Type pi for $\pi$.
The region between $y = x^2$, the $y$-axis and $y = 4$ is rotated about the $y$-axis. Find the volume.
The region between $y = x$ and $y = x^2$ is rotated about the $x$-axis. Find the volume.
The region under $y = \sqrt x$ from $0$ to $4$ is rotated about the $x$-axis. Find the volume.
Find the volume of a sphere of radius $3$.
The region under $y = x$ from $0$ to $3$ is rotated about the $x$-axis, making a cone of radius $3$ and height $3$. Find its volume.
Rotating about the $y$-axis uses:
For the region between $y_1$ (outer) and $y_2$ (inner), the integrand is: