The arc length of y = x^(3/2) from x = 0 to x = 4 is given by:

Question

Accepted Answer

∫₀⁴ √(1 + 9x/4) dx. The correct answer is ∫₀⁴ √(1 + 9x/4) dx. The arc length formula is ∫√(1 + [f'(x)]²) dx. Here f(x) = x^(3/2), so f'(x) = (3/2)x^(1/2). Then [f'(x)]² = (9/4)x. The integral becomes ∫₀⁴ √(1 + 9x/4) dx.

The expression ∫₀⁴ √(1 + 3x/2) dx forgets to square the derivative; it uses f'(x) = (3/2)√x directly instead of [f'(x)]² = 9x/4 inside the square root. The expression ∫₀⁴ √(1 + x³) dx uses [f(x)]² = x³ instead of [f'(x)]², confusing the function itself with its derivative. The expression ∫₀⁴ (1 + 9x/4) dx is missing the square root from the arc length formula, which would give a larger and incorrect result.

The arc length formula L = ∫ₐᵇ √(1 + [f'(x)]²) dx appears on the CLEP exam. The key steps are: compute f'(x), square it, add 1, and place everything under a square root inside the integral. Students should be careful to square the derivative, not the function, and to include the square root. Even if the resulting integral is complex, setting it up correctly earns points.

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