Exploring Vector Field Line Integrals: A Sample Final Exam Problem

[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.

 

[Dick]

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.

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

 

[destroyer130]

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

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!

 

[Dick]

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!

View attachment 53941

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?