Question

How do you determine the length of `x=2t^2`, `y=t^3+3t` for t is between [0,2]?

Answer 1

Treat the point `(x(t), y(t))` as a point moving over time `t`.

The linear distance between two points is based on the Pythagorean theorem
and the distance along the line can be specified as
`int_a^b sqrt(((dx)/(dt))^2 + ((dy)/(dt))^2) dt`

For the given parametric equations this reduces to
`int_0^2 sqrt( 9t^4 + 34t^2 +9)` `dt`

I've poked at this on and off for a couple days but have not come up with an anti-derivative;
therefore the best solution I can offer is to approximate the integral using the sum of areas of rectangles (probably how you were initially taught to understand integrals).

Using 40 rectangles with widths of 0.05 each and heights equal to the average of the function values at the left and right edges, I (with my trusty spreadsheet) calculate the value to be approximately
16.20