Questions: Surface Area of Revolution

Question 1

A student calculates surface area of revolution using S = ∫ π(f(x))² dx. What error has she made?

• A. She used the wrong axis of revolution
• B. She computed volume of revolution (disk method) rather than surface area
• C. She forgot to square the derivative term in the arc length element
• D. She should be integrating with respect to y

Answer: (B)

The formula ∫π(f(x))²dx accumulates cross-sectional areas (disks), giving volume. Surface area accumulates the rim of those circles: each thin strip sweeps out a band with area 2π·f(x)·ds, not a filled disk. Volume stacks filled cross-sections; surface area wraps tape around the outside.

Question 2

Why does the surface area formula include the arc length element √(1 + (f'(x))²) dx rather than simply dx?

• A. To account for the curvature of the surface bending away from the axis
• B. Because the strip being rotated is tilted — its actual length along the curve is longer than its horizontal extent
• C. To convert from polar to Cartesian coordinates
• D. Because surface area always requires a second derivative

Answer: (B)

Each infinitesimal strip lies along the curve, not horizontally. Its actual length is ds = √(1 + (f'(x))²)dx — the hypotenuse of a small right triangle with legs dx and f'(x)dx. When rotated, this strip sweeps a band whose width is the true arc-length, not just the horizontal distance.

Question 3

When revolving y = f(x) about the y-axis instead of the x-axis, the arc length element √(1 + (f'(x))²) dx remains unchanged — only the radius term changes.

• T. True
• F. False

Answer: True

The arc length element depends only on the curve's geometry, not on the axis of revolution. For the x-axis, r = f(x); for the y-axis, r = x. Both formulas share the same structure: 2π·(radius)·(arc length element).

Question 4

The surface area formula S = ∫ 2πf(x)√(1 + (f'(x))²) dx gives the same numerical value as the corresponding volume formula, just measured in different units.

• T. True
• F. False

Answer: False

The formulas measure completely different geometric quantities and produce different values. Volume accumulates filled disk areas (πr²dx); surface area accumulates thin ring strips (2πr·ds). A cylinder of radius r and height h has volume πr²h but lateral surface area 2πrh — entirely different formulas and values, regardless of units.

Question 5

Explain why the surface area of revolution formula uses the arc length element rather than a simple dx term, and what would go wrong if you used dx instead.

Model answer: Each strip of curve is tilted at angle arctan(f'(x)) to the horizontal. Its actual length ds is greater than its horizontal projection dx whenever f'(x) ≠ 0. When rotated, the strip sweeps a band whose width corresponds to the arc length, not the horizontal distance. Using dx would undercount the surface area whenever the curve is not horizontal — the same error as measuring a ramp's length by its horizontal span.

The arc length element captures the true geometric length of each strip. Omitting √(1 + (f'(x))²) would give the correct answer only for horizontal lines (f'(x) = 0 everywhere), where ds = dx. For any other curve, the result would be systematically too small.