Green's Theorem and Lagrange Multipliers for Finding Extrema on a Disc

[helpm3pl3ase]

Hey hows it going??

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.

 

[exk]

http://en.wikipedia.org/wiki/Lagrange_multiplier

the examples section shows how to approach such a problem.

your constraint is the disk so set up f(x,y)-λg(x,y)=Λ(x,y,λ)