velocity and position-ant crawling along y-axis

[Sendell]

An ant is crawling along the y-axis with the velocity (t^2-2)e^(t^2). what is the total distance traveled by the ant from t = 0 to t = 2?
Now, my first impulse was to integrate the velocity function from 0 to 2 and give that as my answer, but since the ant is crawling along the y-axis, I don't think that integrating with x values would be right. I then tried taking the antiderivative of the function, to get the position function. However, my attempts at integrating the velocity function have been unsuccessful.
Where should I go from here?

 

[skeeter]

velocity is v = (t² - 2)e^t²

for 0 < t < 2, v < 0, and v = 0 at t = 2. So, save for the endpoint, the bug is crawling down the y-axis (negative direction) until t = 2. The bug's displacement will be negative.

distance traveled, a scalar quantity, will be \(\displaystyle \L -\int_0^2 v(t) dt\)

I do not think that this velocity function has an elementary, closed-form antiderivative. Unless someone can prove me wrong, I would do a numerical solution using technology to find distance traveled.

 

[daon]

Velocity is zero at t=\(\displaystyle \sqrt{2}\). The bug will slow down from 0 to \(\displaystyle sqrt{2}\) and go backwards (v<0 to v>0) from then until t=2.