Consider the curve defined by and for

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

Problem

Consider the curve defined by

x=t^{2}

and

y=\dfrac{t^{3}}{3}

for

0\le t \le 2

. Which of the following definite integrals gives the arc length of the curve on this interval? (a)

\displaystyle \int_{0}^{2} \sqrt{1+\dfrac{t^{2}}{4}}\,dt

(b)

\displaystyle \int_{0}^{2} \sqrt{4t^{2}+t^{4}}\,dt

(c)

\displaystyle \int_{0}^{2} (4t^{2}+t^{4})\,dt

(d)

\displaystyle \int_{0}^{2} \sqrt{(2t)+(t^{2})}\,dt