Rotating a region about a line sweeps out a solid. Slicing it perpendicular to that line gives disks or washers.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The disk method
Each slice is a circle of radius equal to the function value. Its area is πr², and integrating gives the volume.
The washer method
When the region does not touch the axis, each slice has a hole. Subtract the inner area: π(R² − r²).
Squares first, then subtract
π(R² − r²) is not π(R − r)². Squaring the difference is the single most common error here.
Choosing the variable
Rotating about a horizontal line means slicing vertically and integrating in x. About a vertical line, integrate in y.
Shifted axes
Rotating about y = 2 rather than the x-axis makes the radius f(x) − 2. The radius is a distance, always measured to the axis.
The disc method
Rotating a region about an axis sweeps out circular cross sections. Each disc has area πr², and integrating that area along the axis gives the volume. The radius is the distance from the axis to the curve.
The washer method
When the region does not touch the axis, each cross section is an annulus: π(R² − r²), outer radius squared minus inner. Squaring the difference instead of subtracting the squares is the classic error.
Known cross sections
A solid with square, semicircular or triangular cross sections is built the same way: write the area of one cross section as a function of position and integrate. Rotation is just the case where they are circles.
Sketch the region and one slice
Draw the region, the axis of rotation, and a representative slice with its radius labelled. Almost every error in volume problems is a wrong radius, and the sketch is what catches it.
Step 2: Try It Yourself
Tap and try it out.
- Cubes in one layer4 × 3 = 12
- Layers2
- 4 × 3 × 224 cubic units
Counting every cube gives 24, and so does 4 × 3 × 2. The formula is a shortcut for the count.
Step 3: Watch an Example
One step at a time.
Watch Elena Set Up a Washer
Elena rotates the region between y = x² and y = x, from 0 to 1, about the x-axis.
- Step 1
On (0, 1) the line y = x is above the parabola, so R = x and r = x².
Step 4: Your Turn
Practice makes it stick.
The Disk
Problem 1 of 2
A disk has radius 3. What is its area divided by π?
The Washer
Problem 2 of 2
Outer radius 5, inner radius 3. What is the washer area divided by π?
Slice the Solid
1 of 8
A disk of radius 4. Area divided by π?
2 of 8
Outer radius 6, inner radius 2. Washer area divided by π?
3 of 8
Is π(R² − r²) the same as π(R − r)²? 1 yes, 0 no.
4 of 8
Rotating about the x-axis, which variable do you integrate in? 1 x, 2 y.
5 of 8
Rotating about the y-axis, which variable? 1 x, 2 y.
6 of 8
Rotating y = f(x) about the line y = 2. The radius is f(x) minus what?
7 of 8
Sort each situation by which method it needs.
Tap something to move it.
- Empty
- Empty
8 of 8
Outer radius 10, inner radius 6. Washer area divided by π?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Outer radius 4, inner radius 3. Washer area divided by π?
Question 2 of 2
What is the most common error in the washer method?
What You Learned
- Rotating a region sweeps out a solid, sliced into disks or washers.
- A disk has area πr²; a washer has π(R² − r²).
- Square the radii first, then subtract.