The word document below should explain the problem

Please help, so far your explinations have been better than the professors.

The semester is almost over.

http://i7.photobucket.com/albums/y28...lastscan-1.jpg

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- Nov 20th 2006, 05:33 PMRanger SVOGreens Theorem Help
The word document below should explain the problem

Please help, so far your explinations have been better than the professors.

The semester is almost over.

http://i7.photobucket.com/albums/y28...lastscan-1.jpg - Nov 20th 2006, 06:16 PMThePerfectHacker
What course is this :confused:

(Calculus III does not teach this).

The problem asks**by Green's Theorem**as opposed to the standard method of calculating the the line integral.

You need to find,

$\displaystyle \oint_C 2xy\bold{i}+2x^2\bold{j} \cdot d\bold{R}$

Another notation for this (if that confuses you) is,

$\displaystyle \oint_C 2xydx+2x^2dy$

So the vector field is,

$\displaystyle \bold{F}(x,y)=3xy\bold{i}+2x^2\bold{j}$

Let,

$\displaystyle M(x,y)=3xy$ and $\displaystyle N(x,y)=2x^2$

By Green's theorem,

$\displaystyle \int_D \int \left( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} \right) \, dA$

Where,

$\displaystyle \frac{\partial N}{\partial x}=4x$

And,

$\displaystyle \frac{\partial M}{\partial y}=3x$

So the double integral over that closed region is,

$\displaystyle \int_D \int x\, dA$

Can you take it from there?

(Hint: Divide your integral into two regions $\displaystyle D_1,D_2$ one is the square and the other is a semicircle.)

----

(For Engineering Students)

There happens to be a cool way to get that value if you ever taken an engineering course. You should recognize that is the*first-moment*. And hence you need to find the centroid of this shape. Which is simple for it is one of the standard shapes used. - Nov 21st 2006, 04:51 PMRanger SVO
Ok, so why does he (the professor) have us parametrizing each of the paths and then intergrating them individually and adding them together.

I am asking because what you have shown makes sense and its simpler.

As always I thank you for taking time to answer my questions it is really appreciated. - Nov 21st 2006, 05:24 PMThePerfectHacker
I think your professor wanted to do this problem in two ways. One, the standard way of parametrizing the

*simply connected positevly oriented piecewise smooth closed curve*(this is one of my favorite expressions in math, just look how long it is and how cool it sounds). Second, by using Green's theorem because this is a*simply connected positevly oriented piecewise smooth closed curve*. And show that you get the same result in the end.