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

##### 1 Answer

Treat the point

The linear distance between two points is based on the Pythagorean theorem

and the distance along the line can be specified as

For the given parametric equations this reduces to

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