Mathematics

Problem

For a smooth curve defined parametrically by

x=x(t)

and

y=y(t)

on the interval

[a,b]

, which of the following definite integrals equals the arc length of the curve? a)

\displaystyle \int_{a}^{b}\Bigl[\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}\Bigr]dt

\displaystyle \int_{a}^{b}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dt

\displaystyle \int_{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\,dt

\displaystyle \sqrt{\left(\int_{a}^{b}\frac{dx}{dt}\,dt\right)^{2}+\left(\int_{a}^{b}\frac{dy}{dt}\,dt\right)^{2}}