For a smooth planar curve defined by and with

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

Problem

For a smooth planar curve defined by

x=f(t)

and

y=g(t)

with

t\ge 0

, the arc–length function

s(t)

measured from

t=0

is given by which of the following expressions? (a)

\displaystyle \sqrt{\int_{0}^{t}\bigl[(f'(\tau))^{2}+(g'(\tau))^{2}\bigr] \,d\tau}

(b)

\displaystyle \int_{0}^{t}\bigl[(f'(\tau))^{2}+(g'(\tau))^{2}\bigr] \,d\tau

(c)

\displaystyle \int_{0}^{t}\sqrt{\bigl(f'(\tau)\bigr)^{2}+\bigl(g'(\tau)\bigr)^{2}}\,d\tau

(d)

\displaystyle \sqrt{\bigl(f'(t)\bigr)^{2}+\bigl(g'(t)\bigr)^{2}}