Volumes of revolution

What you are looking at

When the region between $y = f(x)$, the $x$-axis and the lines $x = a$ and $x = b$ is rotated through $2\pi$ about the $x$-axis, each point of the curve traces a circle of radius $y$. A thin slice of width $\Delta x$ becomes a disc of volume $\pi y^2\,\Delta x$. The Sweep button carries the shaded region once round the axis, and the Discs view builds the solid from $n$ such discs, each using the height of the curve at the middle of its slice.

Adding the discs gives $\sum \pi r_i^2\,\Delta x$, and as $\Delta x \to 0$ this sum becomes the integral $V = \pi\int_a^b y^2\,dx$. The working panel evaluates the exact integral step by step and sets it beside the disc total and the difference between them, so you can watch the gap close as $n$ rises. Choosing Around the y-axis relabels the curve as $x = f(y)$ and uses $V = \pi\int_a^b x^2\,dy$ instead.

Try this

1. 1Keep the default curve $y = \sqrt{x}$ from $0$ to $4$. The exact volume is $8\pi \approx 25.1$, and the discs give the same total for every $n$: each disc uses the midpoint radius, and $y^2 = x$ is linear, so nothing is lost.
2. 2Switch to the line $y = \dfrac{x}{2}$, which sweeps out a cone. Set $n = 1$ and then raise it: the disc total starts at $4\pi$, below the exact $\dfrac{16\pi}{3}$, and climbs towards it.
3. 3Press Sweep to watch the flat region travel once round the axis, then drag the picture to tilt and spin the solid. Switch between Discs and Solid to compare the stepped discs with the smooth surface.
4. 4Choose the sine curve on $[0, \pi]$ and read the panel: $V = \pi\int_0^{\pi}\sin^2 x\,dx = \dfrac{\pi^2}{2} \approx 4.93$. Then move $a$ and $b$ and see how the limits change the volume.

Mistakes students make

• Forgetting to square. The integrand is $y^2$, not $y$: for $y = \sqrt{x}$ on $[0, 4]$ the volume is $\pi\int_0^4 x\,dx = 8\pi$, not $\pi\int_0^4 \sqrt{x}\,dx$.
• Dropping the $\pi$ or writing $2\pi$ in front. Each disc is a cylinder of volume $\pi r^2\,\Delta x$, so the factor is exactly $\pi$.
• Rotating about the $y$-axis but still integrating with respect to $x$. About the $y$-axis, write $x$ in terms of $y$, take the limits on the $y$-axis and use $V = \pi\int_a^b x^2\,dy$.

Where it is taught

• IB Maths AA HL 5.17 · Volumes of revolution about the x-axis or y-axis
• AP Calculus AB/BC Unit 8 · Volume with the disc method, revolving around the x- or y-axis (8.9)