How do I set up these line integrals?
First problem:
Line integral of y^3dx + x^2dy where C is the arc of the parabola x=1-y^2 from (0,-1) to (0,1).
Second problem:
Line integral of ydx + (x+y^2)dy where C is the ellipse 4x^2 + 9y^2 = 36, with counterclockwise rotation
Thanks this helps a lot. I knew there was a way to convert a line integral in that form to a double integral but could not find it in my notes.
Edit:
I still need help. I've found my notes on green's theorem but there are no examples on how to apply it to ellipse. I tried turning problem 2 it into a double integral and came up with the double integral of 0. What am I doing wrong. I know I'm probably overthinking this.
Have you not had any instruction in this? Since x= 1- y^2, dx= -2ydy so y^3dx+ x^2dy= y^3(-2ydy)+ (1- y^2)^2dy= -2y^4dy+ (1- 2y^2+ y^4)dy= (-y^4- 2y^2+ 1)dy. Since y is going from -1, to 1, that integral is .
This is a little harder because neither x nor y is a function of the other so we have to set up a "parameterization". Fortunately, there is a "standard" parameterization of an ellipse like this: x= 3cos(t), y= 2sin(t). Do you see why? 4x^2+ 9y^2= 4(9cos^2(t))+ 9(4sin^2(t))= 36(cos^2(t)+ sin^2(t))= 36.Second problem:
Line integral of ydx + (x+y^2)dy where C is the ellipse 4x^2 + 9y^2 = 36, with counterclockwise rotation
Now, from x= 3cos(t), dx= -3 sin(t)dt and from y= 2sin(t), dy= 2cos(t)dt. Going completely around the ellpse, t goes from 0 to .
Thanks that helps me understand it better. I have had instruction in this but I was a more general lecture. We only worked a few examples but it wasn't enough for me to fully grasp how to work any problem. The best way for me to learn anything is to do it repetitively and to do different kinds of examples.
If you use the parameterisation given by HallsOfIvy, with and with , then
If you use Green's Theorem
So
We have the ellipse . If we use the Change of Variables and , the Jacobian is
and the ellipse transforms to the circle given by
and the integral gets transformed to
And the region, now being a circle, should be transformed to polar coordinates. When this happens, we have the bounds and , which gives
The original question was about doing the line integrals. Yes, the itegral around a closed curve can be done as a double integral but I don't think that was what was being asked.
Oh, and when you have , no matter how complicated A is, I hardly think you need to convert to a simpler region! 0 has integral 0 over any region!