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y=y(t), x=x(t)

A=[inte]ydx=[inte]yx'(t)dt

That result seems straighforward.

I also thought, what if I let the VVF

**v**=<x(t),y(t)> represent the same curve. To find the area, under the curve (I have in the back of my mind the concept of velocity and position), I would solve the integral,

**r**=[inte]

**v**dt.

Should these two results be related? I think they should, but the math shows they aren't. Should the magnitidue of the latter equal the absolute value of the former? Looks like no. Why not?