I Help with a 3D Line Integral Problem (segmented line)

[SK97]

TL;DR Summary

Line integral problem involving 3 segmented lines along the xyz plane.
Need some help on how to begin to tackle the problem.

Hi all,

I'm finding it difficult to start this line integral problem.

I have watched a lot of videos regarding line integrals but none have 3 line segments in 3D.

If someone can please point me in the right direction, it would help a lot.

I've put down the following in my workings:

C1: (0,0,0) - (1,0,0)

C2: (1,0,0) - (1,1,0)

C3: (1,1,0) - (1,1,1,1)

How do I proceed from here?

Thank you in advance!

 

[anuttarasammyak]

On $C_1$ dz=dy=0, so only $\int dz$ survives.
On $C_2$ and $C_3$, similarly you should make one integral out of the three.

 

[SK97]

On $C_1$ dz=dy=0, so only $\int dz$ survives.
On $C_2$ and $C_3$, similarly you should make one integral out of the three.

Okay, so i form three integrals and combine them?

Also that's where i am confused, when i am doing C1 shouldn't the 3dz go away too as its treated as a constant?

 

[anuttarasammyak]

(correction) only $\int dx$ survives.
So for $C_1$ the integral is $\int_{C_1} x^2z \ dx$ where z is ...

 

[SK97]

(correction) only $\int dx$ survives.
So for $C_1$ the integral is $\int_{C_1} x^2z \ dx$ where z is ...

So for C1 i have the parametrisation as r(t) = <t,0,0>

I got this by doing:
x= t(1) + (1-t)(0) = t

and zeros for the others.

Is this the correct way to evaluate it?

 

[anuttarasammyak]

I got this by doing:
x= t(1) + (1-t)(0) = t

and zeros for the others.

So the integrand $x^2z$ on $C_1$ is ...

 

[SK97]

So the integrand $x^2z$ on $C_1$ is ...

just t^2 dx right?

 

[anuttarasammyak]

\int x^2z \ dx = z\int x^2 dx =..

 

[SK97]

\int x^2z \ dx = z\int x^2 dx =..

so replacing x with t we get t^2 in the integral but since z= 0 do we get 0 for c1?

sorry if I'm understanding incorrectly

 

[anuttarasammyak]

 

[SK97]

oh thank you so much, so similar set up then for the other integrals and then at the end its just addition of them all, correct?

 

[anuttarasammyak]

 

[SK97]

Thanks a lot! hopefully i don't run into other issues but please be around to help out!

will greatly appreciate it :)

 

[SK97]

Thanks a lot! hopefully i don't run into other issues but please be around to help out!

will greatly appreciate it :)

so i finished calculating for all C values and after the addition got to 7/2 as the answer.

does that seem right to you?

 

[etropy]

Seperate it into the sum of 3 intgrals over c1 c2 and c3. Then parametrize each path. Plug everything in, calculate dx dy and dz in Terms of dt, calculate the 3 integrals then. Add them, and you are done.