A particle moves in the plane with parametric equations and for

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

Problem

A particle moves in the plane with parametric equations

x(t)=t^{2}

and

y(t)=t^{3}

for

0\le t\le2

. Which definite integral equals the total distance traveled by the particle on this interval? (a)

\displaystyle\int_{0}^{2}\sqrt{(2t)^{2}+(3t^{2})^{2}}\,dt

(b)

\displaystyle\int_{0}^{2}\sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}}\,dt

(c)

\displaystyle\int_{0}^{2}\bigl((2t)^{2}+(3t^{2})^{2}\bigr)\,dt

(d)

\displaystyle\int_{0}^{2}\sqrt{(2t)^{2}+(3t)^{2}}\,dt