Question

Verify that the following integral is path independent then find the potential function and use it to identify the integral
\[\int\limits_{(-2,3)}^{(4,2)}(3x^2y-y^2)dx+(x^3-2xy)dy\]

Answer 1

The integral is path-independent if the given vector field is conservative.

To show that a vector field \({\bf F}=\langle M(x,y),N(x,y)\rangle\) is conservative, you have to show that
\[\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}\]

Answer 2

what am I considering to be M & N?

Answer 3

The convention is that a line integral can be written in the form
\[\int_C {\bf F}\cdot d{\bf r}=\int_C M(x,y)~dx+N(x,y)~dy\]

Answer 4

okay, I wasn't familiar with this form. So they are path independent\[M _{y}=3x^2-2y=N _{x}\]

Answer 5

Right, so now you can compute the integral by finding the potential function \(\bar{f}\) and evaluating \(\bar{f}(4,2)-\bar{f}(-2,3)\) using the FTC for line integrals.

Answer 6

alright, I got 118. Looks good. Thanks for your help. I hadn't encountered a problem put that way before

Answer 7

yw