Solving a Complex Integral Problem by Hand

[p0tat0phun]

Homework Statement

The problem is given in the following image.
http://img46.imageshack.us/img46/2972/lskfjsf.png

Homework Equations

∫h(r)*dr = ∫h[r(u)]*r'(u)du

The Attempt at a Solution

I was able to figure out plugging in (t^2+3) at every x and sin(1/2πt) at every y. I then set up the dot product of the substituted equation and r'(u)du (2t + 1/2πcos(1/2πt)). The problem is, the integral ends up being much to complex to work by hand since I'm not allowed to use a calculator for this problem. So I was wondering if there was an easier way to work this problem out?

 

[LCKurtz]

Homework Statement

The problem is given in the following image.
http://img46.imageshack.us/img46/2972/lskfjsf.png

Homework Equations

∫h(r)*dr = ∫h[r(u)]*r'(u)du

The Attempt at a Solution

I was able to figure out plugging in (t^2+3) at every x and sin(1/2πt) at every y. I then set up the dot product of the substituted equation and r'(u)du (2t + 1/2πcos(1/2πt)). The problem is, the integral ends up being much to complex to work by hand since I'm not allowed to use a calculator for this problem. So I was wondering if there was an easier way to work this problem out?

Think about why the first part asked you to find f(x,y) whose gradient is the given vector field. Perhaps you can use that f(x,y) somehow, hint, hint.

 

[p0tat0phun]

Okay, so I found f(x,y) to be:
3x + e^x*y^2 + 3e^x*y + 2e^x + y^2
But I'm still confused on where to continue with that.

 

[LCKurtz]

Okay, so I found f(x,y) to be:
3x + e^x*y^2 + 3e^x*y + 2e^x + y^2
But I'm still confused on where to continue with that.

Your text should have a theorem about evaluating line integrals when you have a potential function f(x,y), and that is the whole point of this problem.