Topic 8.12 Notes – Volume with Washer Method: Revolving Around Other Axes

What the Washer Method Around Other Axes Is

When you revolve a region and there’s a hole in the middle, each cross section looks like a washer.

Area of one washer:

$$\pi R^2 - \pi r^2$$

So the total volume is:

$$V = \int \pi\big(R^2 - r^2\big)$$

The difference now is that the axis of rotation is something like $y = 3$ or $x = -2$.
That means radius = distance to that line, not just the function value.

If rotating around a horizontal line $y = b$:

$$V = \int_c^d \pi\big(R(x)^2 - r(x)^2\big)\,dx$$

where each radius is a vertical distance to $y=b$.

If rotating around a vertical line $x = a$:

$$V = \int_c^d \pi\big(R(y)^2 - r(y)^2\big)\,dy$$

where each radius is a horizontal distance to $x=a$.

Distance is everything here.

Geometry First Then Algebra

Before writing an integral, picture what’s happening.

Suppose the region between $y=4-x^2$ and $y=1$ is rotated about $y=5$.

Washer setup for rotation about $y=5$

Notice what the diagram is emphasizing:

• The axis is above the region.
• Radii are measured downward from $y=5$.
• Outer radius goes to the lower curve.
• Inner radius goes to the upper curve.

So:

• $R(x) = 5 - 1 = 4$
• $r(x) = 5 - (4 - x^2)$

Even though $4-x^2$ is the “top” curve, it creates the inner radius because it’s closer to the axis.

That switch trips people up constantly.

Choosing the Variable

The slices must be perpendicular to the axis.

• Horizontal axis → vertical slices → integrate dx
• Vertical axis → horizontal slices → integrate dy

If rotating around $x = 2$, your radii must look like:

• right function − 2
• left function − 2

And you’ll integrate with respect to y, which may require rewriting equations.

On quizzes and FRQs, using the wrong variable usually costs multiple setup points.

Full Setup Checklist

When you’re staring at one of these:

1. Sketch the region and the axis.
2. Decide dx or dy (perpendicular slices).
3. Find intersection points for bounds.
4. Write outer radius as a distance.
5. Write inner radius as a distance.
6. Plug into $\int \pi(R^2 - r^2)$.

Keep the subtraction inside the square:

$$( f(x) - b )^2$$

If the axis is negative, be careful:

• Rotating about $y=-3$ means subtracting $-3$, which becomes adding 3.

Sign mistakes are the most common algebra error here.

Disk vs Washer Quick Comparison

If there’s empty space between the region and the axis, it’s a washer.

Common Mistakes That Cost Points

• Writing $\pi(R-r)^2$. That’s wrong. It must be $\pi R^2 - \pi r^2$.
• Forgetting to adjust for the axis line.
• Picking the “top” function as outer radius instead of checking distance.
• Using dx when the axis is vertical.
• Expanding everything too early and making algebra errors.

On AP FRQs, most volume points come from the correct integral setup, not from grinding out the final number. A clean, correct integral with clear radii earns major credit.