Which of the following integrals gives the exact length of a smooth curve traced once…

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

Problem

Which of the following integrals gives the exact length of a smooth curve traced once by the vector-valued function

\mathbf r(t)

for

t\in[a,b]

? (a)

\displaystyle \int_{a}^{b} \lvert\mathbf r(t)\rvert\,dt

(b)

\displaystyle \int_{a}^{b} \lvert\mathbf r'(t)\rvert\,dt

(c)

\displaystyle \int_{a}^{b} \mathbf r'(t)\,dt

(d)

\displaystyle \lvert\mathbf r(b)-\mathbf r(a)\rvert