Which integral represents the exact length of a smooth curve given by on the interval ?

MATHEMATICS · Single correct · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

Which integral represents the exact length of a smooth curve given by y=f(x)y=f(x) on the interval [a,b][a,b]? (a) ∫ab(1+(f′(x))2) dx\displaystyle\int_{a}^{b}\bigl(1+(f'(x))^{2}\bigr)\,dx (b) ∫ab1+(dxdy)2 dy\displaystyle\int_{a}^{b}\sqrt{1+\bigl(\tfrac{dx}{dy}\bigr)^{2}}\,dy (c) ∫ab1+(f′(x))2 dx\displaystyle\int_{a}^{b}\sqrt{1+\bigl(f'(x)\bigr)^{2}}\,dx (d) ∫ab(f′(x))2 dx\displaystyle\int_{a}^{b}\sqrt{\bigl(f'(x)\bigr)^{2}}\,dx

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