Calculus of Several Variables: Integration

[Storm Butler]

Homework Statement

The problem is int(y^3 dx-x^3dy) with the region as x^2+y^2<4 and c equal to the boundary. we are supposed to integrate along C+. This may be unclear since I don't know how to type it up in latex however here is a link to the HW assignment. The problem is problem number 1.

http://www.math.vt.edu/people/sun/class_policy//home_n8.pdf

Homework Equations

So far in class we have been doing these surface integrals and line integrals ect. To be honest i don't get what the teacher is really telling us. It seems like we parameterize the boundary and the vector or scalar function we are integrating and then we find the normal vector and multiply by that and then integrate but I am really lost for this. Also i don't know what the C+ part means. Does it mean to integrate just the top half of the cirlce or something?

Thanks for any help.

The Attempt at a Solution

 

[grindfreak]

Yeah I'm pretty sure he's talking about the upper half of the boundary, I really don't know what else it could be since that is obviously a line integral. Anyway, with that in mind x^2 + y^2 = 4 since it's on the boundary. Then all you have to do is find y in terms of x and dx in terms of dy (or vice versa) and integrate over whatever variable you chose.

 

[DryRun]

In your problem, it says, let C = ∂D, so first, you need to find C, which is the total differential of D.

Recall that the total differential is the sum of the partial differentials. You will get the equation of C. Draw a graph to understand what region is concerned.

I got the answer: 0 but you'll obviously have to show your work and find the limits of line integral.

Waiting until the eve of returning an assignment is always a bad living principle, but you already know that.

 

[Storm Butler]

Ya I know, it's been a busy week. Anyways I was reading through my notes a little more and i didn't really see what you were talking about sharky but i tried again and this is what i did. I turned the integral into a double integral using Greens theorem. So i have it becoming doubleint(-3x^2-3y^2)dxdy with the y-boundaries as sqrt(4-x^2) and -sqrt(4-x^2) and the x-boundaries as -2 to 2. Does this make sense? I looked around and i figured that the C+ just meant I should use the positive orientation.

 

[Storm Butler]

If you guys are still willing to help, i believe i figured out how to do 3 with the divergence thm. Number two however I am trying to use stokes. Here's the problem though, the integral should turn into the closed integral over the ellipse that bound the ellipsoid (i think) so i get int(2ydx+xzdy+sin(x^2+y^4)dz) then i tried parameterizing this into x=cos(t) y=sqrt(2)sin(t) to get the ellipse boundary (x^2+1/sqrt(2)y^2=1) the problem is the sin(x^2+y^4) term then becomes something horrible that i can't integrate.

 

[Ray Vickson]

In your problem, it says, let C = ∂D, so first, you need to find C, which is the total differential of D.

Recall that the total differential is the sum of the partial differentials. You will get the equation of C. Draw a graph to understand what region is concerned.

I got the answer: 0 but you'll obviously have to show your work and find the limits of line integral.

Waiting until the eve of returning an assignment is always a bad living principle, but you already know that.

I think you are interpreting things incorrectly. Sometimes the notation ∂ is used as a shorthand to denote the boundary of a set, so C = ∂D says "C is the boundary of D". It has nothing to do with partial derivatives. The only remaining issue is what is meant by C⁺, but presumably that was defined somewhere in the textbook or course notes.

RGV