So I was given the following prompt:
"The position of a car moving along a flat surface at time $t$ is modeled by $(x(t), y(t))$ with velocity vector $v(t)=<3+6\sin(3t), 1+e^{2t}>$ for $0 \leq t \leq 2$. Both $x(t)$ and $y(t)$ are measured in feet, and $t$ is measured in seconds. At time $t=0$, the car is at position $(0,0)$. Write, but do not evaluate, an integral expression that gives the total distance traveled by the car from time $t=0$ to time $t=2$."
I guess I'm a bit confused about what that integral might look like. I understand that the bounds of the integral would be from $0$ to $2$, but I'm a bit confused about what expression the question is looking for to be integrated. Any clarification would be appreciated!
Best Answer
Since $\mathbf{v}(t) = \langle \frac{dx}{dt}(t), \frac{dy}{dt}(t)\rangle$ and you were given $\mathbf{v}(t)$, you should have enough to plug into the arc length formula now.