Welcome to the Washer Method!
In our previous look at the Disc Method, we learned how to find the volume of a solid object created by spinning a region flush against an axis. But what happens if there is a gap between the region and the line we are spinning it around? We get a solid with a hole in the middle—kind of like a donut, a pipe, or a literal hardware washer!
The Washer Method is simply an extension of the Disc Method that accounts for that empty space in the center. If you can find the area between two curves, you are already halfway to mastering this chapter.
1. What is a "Washer" Anyway?
Imagine a circular disc with a smaller circle cut out of its center. In geometry, the area of this shape is the area of the large outer circle minus the area of the small inner circle:
\( \text{Area of Washer} = \pi(R)^2 - \pi(r)^2 \)
Or, more commonly written as: \( \pi(R^2 - r^2) \)
Where:
\( R \) (Capital R) is the Outer Radius (the distance from the axis to the outer curve).
\( r \) (Lowercase r) is the Inner Radius (the distance from the axis to the inner curve).
2. The General Formulas
When we rotate a region bounded by two functions around an axis, we sum up an infinite number of these thin washers using an integral.
Revolving Around the x-axis (or any horizontal line)
If we are revolving around the \( x \)-axis, our radii are vertical distances, so we use functions of \( x \) and integrate with respect to \( x \):
\( V = \pi \int_{a}^{b} ([R(x)]^2 - [r(x)]^2) \, dx \)
Revolving Around the y-axis (or any vertical line)
If we are revolving around the \( y \)-axis, our radii are horizontal distances, so we use functions of \( y \) and integrate with respect to \( y \):
\( V = \pi \int_{c}^{d} ([R(y)]^2 - [r(y)]^2) \, dy \)
Key Takeaway: Always remember the \( \pi \) outside the integral! It is the most common thing students forget on the AP Exam.
3. Determining "Outer" vs. "Inner" Radius
Don't worry if this seems confusing at first! A simple sketch is your best friend here.
Step 1: Draw the region and the axis of revolution.
Step 2: Draw a representative rectangle or a "radius line" from the axis of revolution into the shaded region.
Step 3: The curve the line hits last (the one further away) is your Outer Radius \( R \).
Step 4: The curve the line hits first (the one closer to the axis) is your Inner Radius \( r \).
Analogy: Think of the axis of revolution as the "center of the world." The further you travel away from the center, the "Outer" you get.
4. Revolving Around Other Axes (Topic 8.12)
The AP Exam often asks you to rotate around lines other than the \( x \) or \( y \) axes, such as \( y = 3 \) or \( x = -2 \). The logic remains exactly the same, but our radii formulas change slightly.
Horizontal Axis \( y = k \)
\( R(x) = |\text{Outer Curve} - k| \)
\( r(x) = |\text{Inner Curve} - k| \)
Example: If you rotate the region between \( y = x^2 \) and \( y = 4 \) around the line \( y = 5 \), the outer radius \( R \) is the distance from \( 5 \) to \( x^2 \), and the inner radius \( r \) is the distance from \( 5 \) to \( 4 \).
Vertical Axis \( x = h \)
\( R(y) = |\text{Outer Curve} - h| \)
\( r(y) = |\text{Inner Curve} - h| \)
Quick Tip: For radii, always use (Top - Bottom) for horizontal axes and (Right - Left) for vertical axes to keep your distances positive!
5. Step-by-Step Problem Solving
Follow these steps to avoid getting lost in the algebra:
- Find the Intersection Points: Set the functions equal to each other to find your limits of integration (\( a \) and \( b \)).
- Identify the Radii: Determine which function is \( R(x) \) and which is \( r(x) \) by looking at the graph.
- Set up the Integral: Write out the full expression: \( \pi \int_{a}^{b} (R^2 - r^2) dx \).
- Square FIRST, then Subtract: This is the "Algebra Danger Zone." (See "Common Mistakes" below).
- Evaluate: Use the Fundamental Theorem of Calculus or your graphing calculator if it's a calculator-active question.
6. Common Mistakes to Avoid
The "Quantity Squared" Error:
Students often accidentally write \( (R - r)^2 \). This is WRONG.
The correct form is \( R^2 - r^2 \).
Memory Aid: You are subtracting two separate circular areas, so you need two separate squares!
Forgetting the \( \pi \):
The formula is based on the area of a circle, \( \pi r^2 \). If you leave out the \( \pi \), you're just finding a weird sum of squares, not a volume.
Wrong Variables:
If you revolve around a horizontal line (like the \( x \)-axis), everything must be in terms of \( x \). If you revolve around a vertical line (like the \( y \)-axis), everything must be in terms of \( y \).
7. Summary Table
Axis of Revolution: Horizontal (\( x \)-axis or \( y = k \))
Integral Setup: \( dx \)
Radii: Vertical distances (Top - Bottom)
Axis of Revolution: Vertical (\( y \)-axis or \( x = h \))
Integral Setup: \( dy \)
Radii: Horizontal distances (Right - Left)
Key Takeaway
The Washer Method is just Big Circle minus Little Circle. If you can identify the distance from the axis of revolution to each boundary, you can set up any volume problem with confidence!