Mathematics

Problem

For a curve defined parametrically by

x(t)

and

y(t)

a\le t\le b

, which of the following integrals gives the length of the curve on that interval? a)

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

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

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

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