For a plane 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 plane curve defined parametrically by

x=f(t)

and

y=g(t)

on the interval

a \le t \le b

, which of the following definite integrals would compute the length of the curve? (a)

\displaystyle \int_{a}^{b}\left(\bigl[f'(t)\bigr]^{2}+\bigl[g'(t)\bigr]^{2}\right)\,dt

(b)

\displaystyle \int_{a}^{b}\sqrt{\bigl[f'(t)\bigr]^{2}+\bigl[g'(t)\bigr]^{2}}\,dt

(c)

\displaystyle \int_{a}^{b}\sqrt{f(t)^{2}+g(t)^{2}}\,dt

(d)

\displaystyle \int_{a}^{b}\bigl(|f'(t)|+|g'(t)|\bigr)\,dt