A region between y = 3 and y = 1 (where both are above the x-axis) is revolved about the x-axis. Which expression gives the area of a washer cross-section?
Aπ(3 − 1)² = 4π
Bπ(3² − 1²) = 8π
Cπ(3 + 1)² = 16π
Dπ · 3² = 9π
The washer is a disk of radius R = 3 with a disk of radius r = 1 removed from its center. Its area is the difference of the two circular areas: πR² − πr² = π(9 − 1) = 8π. The most common error is computing π(R − r)² = π(4) = 4π — this squares the difference rather than taking the difference of squares. π(R−r)² ≠ π(R²−r²) unless r = 0.
Question 2 Multiple Choice
The region between y = x² and y = x (where x² ≤ x for 0 ≤ x ≤ 1) is revolved about the axis y = −1. What are the outer and inner radii for the washer at position x?
AR = x, r = x²
BR = x + 1, r = x² + 1
CR = x − 1, r = x² − 1
DR = 1 − x, r = 1 − x²
The axis is y = −1. The outer radius is the distance from the axis to the farther curve (y = x, the upper curve): R = x − (−1) = x + 1. The inner radius is the distance to the closer curve (y = x²): r = x² − (−1) = x² + 1. When revolving about an axis other than y = 0, you must express each radius as the distance from the axis, not the raw y-value. Forgetting to add 1 (the axis offset) to both radii is the most common setup error.
Question 3 True / False
The washer method can be viewed as computing the volume of a large solid of revolution and subtracting the volume of a smaller solid of revolution that corresponds to the hollow interior.
TTrue
FFalse
Answer: True
This is the conceptual foundation of the washer method. When you revolve the region between two curves, you sweep out the volume the outer curve would create (the full disk) minus the volume the inner curve would create (the hole). Integrating πR²(x) gives the outer solid's volume; subtracting ∫πr²(x) removes the hollow interior. Writing V = ∫π(R² − r²) dx is just doing both in one step.
Question 4 True / False
If f(x) > g(x) > 0 and you revolve the region between them about y = 0, the washer volume formula V = ∫π(f − g)² dx is correct.
TTrue
FFalse
Answer: False
This is the most dangerous misconception in the washer method. The correct formula is V = ∫π(f² − g²) dx, not V = ∫π(f − g)² dx. The difference of squares ≠ the square of the difference: f² − g² = (f+g)(f−g), while (f−g)² = f² − 2fg + g². Using (R−r)² instead of (R²−r²) gives a wrong volume.
Question 5 Short Answer
Explain why, when revolving about the axis y = k (where k ≠ 0), you must adjust the radii rather than using the raw y-values of the boundary curves.
Think about your answer, then reveal below.
Model answer: The radius of each cross-sectional ring must be the perpendicular distance from the axis of revolution to the curve, not the curve's y-value. When the axis is y = k, the distance from the axis to a curve at height y is |y − k|, not y. Using raw y-values only works when k = 0 because then distance = |y − 0| = y (for y > 0).
The formula π(R² − r²) requires R and r to be radii — distances from the center of each circular cross-section to its boundary. The axis of revolution is that center. If the axis is y = 2 and a curve is at y = 5, the ring's outer radius is 5 − 2 = 3, not 5. Drawing the cross-section explicitly and labeling the distances from the axis (not from y = 0) is the reliable way to set up these problems correctly.