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I am having some trouble on this problem:

Use Green's Theorem to evaluate the line integral

∫C F . dr

where F =< y^3 + sin 2x, 2x(y^2) + cos y > and C is the unit circle x^2 + y^2 = 1 which is

oriented counterclockwise.

I started like so:

∫C Pdx + Qdy = ∫∫D Qx - Py dA

Where P = y^3 + sin 2x

Q = 2x(y^3) + cos y

and

Px = 3y^2

Qx = 2y^2

Now we have

∫∫D 2y^2 - 3y^2 dA

= ∫(2π to 0)∫(1 to 0) 2y^2 - 3y^2

Now I am confused on where to go or even if I did this correctly. Please help.

The other problem I had trouble with goes as so:

Find the maximum and minimum values of the function

f(x, y) = x^2 + y^2 - 2x + y

on the disc x^2 + y^2 ≤ 5.

Solution:

fx = 2x - 2 = 0 --> x=1

fy = 2y + 1 = 0 --> y=1/2

Pt(1, 1/2)

Now we use Lagrange Multipler:

(1) fx: 2x - 2 = λ2x

(2) fy: 2y + 1 = λ2y

(3) x^2 + y^2 = 5

From here I know you have to solve for one of the equations then plug in.. I picked (2) to solve for y, but I am not sure how to solve it?? Or even if I approached this right.. Any help is appreciated.. Thank you.