# Integral on end points

1. Jan 24, 2012

### bugatti79

1. The problem statement, all variables and given/known data

Show that the following line integral depends only on the end points of C by using a suitable theorem.

Then evaluate without parameterising it.

2. Relevant equations

$\int_{(0,0)}^{(3,2)} 2xe^y dx + x^2e^y dy$

3. The attempt at a solution

Is this the theorem...

$\displaystyle \int F(x,y) dr =\int_C \frac{\partial \phi}{\partial x} dx +\frac{\partial \phi}{\partial y} dy =\int_a^b (\frac{\partial \phi}{\partial x} \frac{dx}{dt} +\frac{\partial \phi}{\partial y} \frac{dy}{dt}) dt$...........?

2. Jan 24, 2012

### lanedance

do you know about conservative functions and vector fields ?

3. Jan 24, 2012

### bugatti79

Yes, this force vector field is conservative because g_x=f_y, hence is independent of path C..but I dont know what theorem to use?

4. Jan 24, 2012

### lanedance

5. Jan 24, 2012

### bugatti79

It must be simpler than I thought

I calculate the potential function $\phi (x,y)= x^2 e^y +C$. we know the gradient of the potential function is the force vector field given in the question. I integrated backwards to get the potential, s I guess thats the theorem..right?

I evaluate the force field to be 9e^2....based on $\displaystyle \int \nabla \phi \dot dr = \phi(x_1, y_1) - \phi (x_0, y_0)$...

6. Jan 24, 2012

looks good