At time $t\geq 0$, a particle moving in the $xy$-plane has velocity vector given by $v(t)= \Big\langle 4e^{-t}, \sin{(1+\sqrt{t})} \Big\rangle$. What is the total distance the particle travels between $t=1$ and $t=3$ ?

Summary Submit

Recall that when given parametric equations, arc length can be calculated with the following:

$$L = \int\limits_a^b \sqrt{[x'(t)]^2 + [y'(t)]^2}   dt$$

Since we are given the velocity vector $(x'(t) \text{ and } y'(t))$, we can use our calculator to evaluate the following integral.

$$L = \int\limits_1^3 \sqrt{(4e^{-t})^2 + (\sin{(1+\sqrt{t})})^2 }   dt$$ $$L = \int\limits_1^3 \sqrt{16e^{-2t} + \sin^2{(1+\sqrt{t})} }   dt$$ $$= \boxed{1.861}$$