Calculus III problem
Hi, so tomorrow's pretty much the last day of the summer semester for me and I have a homework assignment due tomorrow that's confusing me sooo much..
Im here asking for some help in direction on how to do these problems so that I can find the solution by myself... I would really really appreciate any help anyone could provide ...
I am not sure how to represent the domain as a function, and if anyone could point me in the right direction, I'm sure I could get the answer.
The Problem -
I would really really really appreciate any help at all. I need to know how to represent the domain as a function.
Don't think of the domain as one big function. Think of it as 3 parts as in the descrption:
C1: line from (0,0) to (1,0)
C2: circular arc from (1,0) to the line x=y. What is x in this case?
C3: line from (x,x) to (0,0)
If you don't know how to parametrize lines, look in your book somewhere around parametric equations of a line.
The circular arc you can just parametrize using sines and cosines.
I hope this helps!
I think we should go for Green's Theorem.
So, lets do it:
Let have and .
Now we have to determine the region bounded by the curve C. I've made a sketch:
You see that we're "walking" counter-clockwise - thus we should have . There would be a negative signal if we walked clockwise.
Well. We should determine the limits of integration. I mentioned above the convenience because this integral is asking so much that we change it for polar coordinates - therefore we can easily find the limits of integration (I think the sketch speaks by itself). So we put:
Our angle will vary from 0 to and the radius will vary from 0 to 1 (unit circle). This will lead us to:
Ah, before, we can't forget the Jacobian (I will not show this, but you should know how its done), so . Finally, the integral becomes:
because by our change.
It says counter-clockwise in the question!
Oh, my mistake, sorry. I'll correct it.
Thank you so much for the quick replies. I appreciate it a looooot! I will try it out and see what I can come up with. And keep you guys updated.
Already done, I guess you can follow now.