13.4#3

I can't seem to setup my bounds properly for this problem. Honestly, setting up my bounds for all 
of these Green's Theorem problems has been a struggle. How should I go about setup up the bounds?



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Well, ok, you've got a piece of pie of radius 4. It's 1/8 of the whole pie. *If* you were going to actually calculate the circulation, that would be the path integral along the edge of the piece of pie. Since it has three pieces, you'd have to break up the path into three pieces: the part going outward from the origin along the x-axis, which can be parameterized as r(t) = <4t,0> for 0≤t≤1, the counterclockwise circular part of radius 4 moving from the x-axis upward and to the left to the point (2√2,2√2), which can be parameterized as r(t) = <4 cos(t), 4 sin(t)> for 0≤t≤π/4, and then the radial part moving inward that can be parameterized as r(t) = (1-t)<2√2, 2√2>. You'd have to substitute x(t) and y(t) for each part and then integrate with respect to t and add them all up.

*BUT* you're not going to do it that way, instead you're going to compute
∂F_2/∂x - ∂F_1/∂y = 5-(-4) = 9,
then use ∫∫_D ∂F_2/∂x - ∂F_1/∂y dA = 9 ∫∫_D dA = 9(Area of circle)/8 = 9(π 4^2)/8
=9*π*16/8 = 18π = 56.55

By the way, I changed the due date to tomorrow night.