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

1. Washer Method Around a Horizontal or Vertical Line

When you rotate a region and there’s empty space between the region and the axis, you get washers.

Picture a slice perpendicular to the axis. In the example below, the region is revolved around the vertical line $x = 2$, creating ring-shaped cross sections with an outer and inner radius.

Washer cross sections formed by rotating a region around $x = 2$

The volume comes from adding up areas of these rings:

$$V = \int_a^b \pi\left(R(x)^2 - r(x)^2\right)\,dx$$

• $R(x)$ = outer radius (farther from axis)
• $r(x)$ = inner radius (closer to axis)
• Always outer² − inner²

The radius is always a distance from the curve to the axis of rotation.

Rotating Around a Horizontal Line $y = b$

• Radii are vertical distances
• Use vertical slices
• Integrate with respect to x

$$R(x) = \text{top function} - b$$

$$r(x) = \text{bottom function} - b$$

Rotating Around a Vertical Line $x = a$

• Radii are horizontal distances
• Use horizontal slices
• Integrate with respect to y

$$R(y) = \text{right function} - a$$

$$r(y) = \text{left function} - a$$

Everything comes back to distance from the axis.

2. How to Set Up the Integral Correctly

This is where most points are lost, not in the integration.

Step-by-step setup

1. Sketch the region and axis

   

   Region between $f(x)=\sqrt{x}$ and $g(x)=1$ revolved about the line y = 1

   In the example shown, the region between $f(x)=\sqrt{x}$ and $g(x)=1$ is revolved about the x-axis, producing washers. Seeing the picture makes it clear which curve is farther from the axis.

2. Find intersection points

   • Solve where the curves are equal.
   • These become your limits of integration.

3. Identify outer and inner radii

   • Outer = farther from axis
   • Inner = closer to axis
   • Write them as distances (include subtracting the axis value).

4. Write the formula

   $$V = \int \pi(R^2 - r^2)\,dx \quad \text{or} \quad dy$$

5. Quick check

   • Correct variable?
   • Outer minus inner?
   • Axis subtraction included?
   • Bounds match variable?

On FRQs, a wrong setup usually means zero for that part, even if your algebra is perfect.

3. Choosing dx or dy

You slice perpendicular to the axis of rotation.

If rotating around a vertical line and your functions are given as $y=f(x)$, you may need to rewrite them as $x=g(y)$. That rewrite is often the hint that you should be integrating with respect to $y$.

On no-calculator sections, clean variable choice makes a huge difference.

4. Special Structures You Should Recognize

Rotating Around the x- or y-Axis

That just means $a=0$ or $b=0$.
Example: rotating around $y=0$ gives $R(x)=f(x)$.

Still subtract the axis, even if it’s zero.

Rotating Around a Line Above or Below

If rotating around $y=-3$, then:

$$R(x) = \text{top curve} - (-3)$$

Subtracting a negative becomes addition. This is one of the most common sign mistakes on quizzes.

Disc Case (No Hole)

If the region touches the axis, then:

$$r(x) = 0$$

So volume simplifies to:

$$V = \int_a^b \pi R(x)^2\,dx$$

Same method. Just no inner radius.

5. Common Mistakes

• Writing $(R - r)^2$ instead of $R^2 - r^2$
• Forgetting to subtract the axis value
• Choosing “top” and “bottom” instead of checking distance from the axis
• Using the wrong variable (dx vs dy)
• Getting negative volume because radii were reversed

If your integrand is negative before integrating, something is wrong.