Topic 8.9 Notes – Volume with Disc Method: Revolving Around the x- or y-Axis

What the Disc Method Is

When you rotate a region around an axis, it creates a 3D solid. The disc method works when:

• The region is bounded by one function and the axis of rotation
• The solid has no hole in the middle

If you slice the solid perpendicular to the axis of rotation, each slice is a thin disc.

Each disc has:

• Radius = distance from the curve to the axis
• Thickness = $dx$ or $dy$
• Volume = area of circle × thickness

So one tiny piece has volume:

$$\pi r^2 \, dx \quad \text{or} \quad \pi r^2 \, dy$$

A definite integral adds all those tiny volumes together.

Revolving Around the x-Axis

If the region is rotated around the x-axis, your slices are vertical and you integrate with respect to x.

In the example below, the region under $y=\sqrt{x}$ from $x=0$ to $x=4$ is revolved around the x-axis. A vertical slice becomes a circular disc when rotated.

Disc method around the x-axis

If the function is $y = f(x)$, then:

• Radius = distance from curve to x-axis = $f(x)$
• Bounds are x-values

The volume formula is:

$$V = \int_a^b \pi [f(x)]^2 \, dx$$

Quick Example

Rotate $y = \sqrt{x}$ from $x=0$ to $x=4$ around the x-axis.

$$V = \int_0^4 \pi (\sqrt{x})^2 dx$$

Since $(\sqrt{x})^2 = x$,

$$V = \pi \int_0^4 x \, dx = \pi \left[\frac{x^2}{2}\right]_0^4 = \pi \cdot 8 = 8\pi$$

Notice how the squaring often simplifies nicely.

Revolving Around the y-Axis

Now suppose we rotate around the y-axis.

Your slices must still be perpendicular to the axis, so they are horizontal. That means you integrate with respect to y.

If written as $x = f(y)$, then:

• Radius = distance from curve to y-axis = $f(y)$
• Bounds are y-values

The formula becomes:

$$V = \int_c^d \pi [f(y)]^2 \, dy$$

Important move

If you're given $y = x^2$ but rotating around the y-axis, you must rewrite it:

$$x = \sqrt{y}$$

If your integrand has $x$ in it but you're integrating with respect to $y$, something is wrong.

How to Set It Up Correctly

When you're staring at a problem, walk through this logic:

1. What axis am I rotating around?
   • x-axis → use $dx$
   • y-axis → use $dy$
2. What is the radius?
   • It is always the distance from the curve to the axis.
   • Around x-axis → radius is a y-value.
   • Around y-axis → radius is an x-value.
3. Square the radius. This is the mistake people make most often.
4. Use bounds that match your variable.
   • $dx$ → x-bounds
   • $dy$ → y-bounds

On FRQs, even if you don’t finish the integral, a correct setup earns most of the credit. The π, the square, and the correct bounds all matter.

When Disc Method Works (and When It Doesn’t)

The disc method applies when:

• The region touches the axis of rotation
• There is only one boundary function
• The cross-section is a solid circle

If there’s a gap between the curve and the axis, the cross-section is a ring. That’s the washer method, which is the next topic.

Quick mental check:

• Filled-in circle → disc
• Circle with a hole → washer