Two students are finding the length of the curve and on

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

Problem

Two students are finding the length of the curve

x(t)=t

and

y(t)=t^{2}

0\le t\le1

. Student A:

\displaystyle \int_{0}^{1}\sqrt{1+(2t)^{2}}\,dt

Student B:

\displaystyle \int_{0}^{1}\sqrt{t^{2}+(t^{2})^{2}}\,dt

Identify which student's integral correctly represents the arc length and briefly state why.