For a smooth curve defined parametrically by and on the interval

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

For a smooth curve defined parametrically by x=x(t)x=x(t) and y=y(t)y=y(t) on the interval [a,b][a,b], which of the following definite integrals equals the arc length of the curve? (a) ∫ab[(dxdt)2+(dydt)2]dt\displaystyle \int_{a}^{b}\Bigl[\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}\Bigr]dt (b) ∫ab1+(dydx)2 dt\displaystyle \int_{a}^{b}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dt (c) ∫ab(dxdt)2+(dydt)2 dt\displaystyle \int_{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\,dt (d) (∫abdxdt dt)2+(∫abdydt dt)2\displaystyle \sqrt{\left(\int_{a}^{b}\frac{dx}{dt}\,dt\right)^{2}+\left(\int_{a}^{b}\frac{dy}{dt}\,dt\right)^{2}}

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