Which of the following is the arc length formula for y = f(x) from a to b?

• A. ∫_a^b sqrt(1 + (f'(x))^2) dx
• B. ∫_a^b sqrt(1 + f'(x)) dx
• C. ∫_a^b (1 + (f'(x))^2) dx
• D. ∫_a^b sqrt((f'(x))^2) dx

Answer: (A)

The arc length of a curve comes from summing tiny straight segments. For a function y = f(x), a tiny horizontal change dx and vertical change dy form a small piece ds = sqrt(dx^2 + dy^2). Since dy = f′(x) dx, this becomes ds = sqrt(1 + (f′(x))^2) dx. Integrating these tiny lengths from a to b gives the arc length: ∫ from a to b of sqrt(1 + (f′(x))^2) dx.

Why the others don’t fit: removing the square root or dropping the square changes the geometry; for instance, sqrt(1 + f′(x)) isn’t derived from ds = sqrt(dx^2 + dy^2) and isn’t the actual infinitesimal length. Likewise, integrating 1 + (f′(x))^2 ignores the necessary square root, misrepresenting the true length, and taking sqrt((f′(x))^2) gives |f′(x)|, missing the horizontal base dx and the 1 term under the radical.