Direct Integration vs. Green's Theorem

[wifi]

Problem:

$\int_R (x-y)dx \ dy=-2/3$ for $R=\{(x,y):x^2+y^2 \geq 1; y \geq 0\}$ by

a.) Direction integration,

b.) Green's theorem.

Attempt at a Solution:

I'm a little confused with part a. Wouldn't the region R be defined by all the points above the y-axis that lie on, in addition to above, the circle of radius 1 centered at the origin?

I'm confused on what the limits of integration would be for the integral $\iint (x-y) dx \ dy$.

 

[vela]

I'm sure that's a typo. It probably should be $x^2+y^2 \le 1$.

 

[wifi]

I'm sure that's a typo. It probably should be $x^2+y^2 \le 1$.

So by direct integration I have,

$$\int_{-1}^{1} \int_{0}^{\sqrt{1-x^2}}(x-y) dy \ dx = 2/3$$ Is it supposed to be negative?

 

[D H]

You made a mistake somewhere. It is -2/3 rather than +2/3.

 

[wifi]

Dang. My limits are correct, yes?

Also, how could I apply Green's theorem. I thought the idea was to change a line integral into a double integral, right?

EDIT: Never mind, it was an error on my part. Still not sure how to apply Green's theorem.

 

[vela]

Well, you have a double integral. You want to change it into a line integral.

 

[wifi]

Well, you have a double integral. You want to change it into a line integral.

So it'd just be a line integral over the boundary of the region, correct?

Since Green's theorem in a plane is given by $$\iint_R (\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y})dx \ dy \ =\oint_C (F_1 dx +F_2 dy)$$

for this specific problem we'd have $\frac{\partial F_2}{\partial x}=x$ and $\frac{\partial F_1}{\partial y}=y$.

Thus, $F_1=\frac{1}{2}y^2$ and $F_2=\frac{1}{2}x^2$. Right?

 

[vela]

Looks good.