# Line integral of Vector field

1. Dec 14, 2012

### destroyer130

This problem is about Line integral of Vector Field. I believe the equation i need to use is:

$\int$F.dr = $\int$F.r'dt, with r = r(t)

I try to solve it like this:
C1: r1= < 1 - t , 3t , 0 >
C2: r2= < 0 , 3 - 3t , t >
C3: r3= < t , 0 , 1 - t >

After some computation, I got stuck at the part that have 2 Gaussian Integrals!

$\int$(t from 0 -> 1) [-3t + 3t^2 + e^(t^2) - e^[(t-1)^2]]dt

I see the answer is 1/2. I check my integrals and observe somehow these 2 Gaussian either cancel out or both equals 0, but I just have no clue how to show it. Another idea I could think of is that there is other way to solve this problem without involving doing those integrals.

Thanks for checking out my problem.

2. Dec 14, 2012

### Dick

Yes, you can show they cancel. Take the integral of e^[(1-t)^2] and apply the substitution u=1-t.

3. Dec 14, 2012

### destroyer130

Wow i didn't know that there's such technique. This is from my sample final exam about Vector Integral. Could you look at the problem i attached and tell me if there's any other way that didn't have to go through that Gaussian Integrals? Thanks a lot Dick!

4. Dec 14, 2012

### Dick

It's a trick you can use to show some definite integrals are related. It's not much of a general technique. Why not apply Stoke's theorem?