Green's Theorem?

Question

Use Green's Theorem to evaluate the given line integral.

∮_C 2xydx+y^2dy, where C is the closed curve formed by y=x/2 and y=sqrt (x) between (0,0) and (4,2)

Answer 1

You will recall green's theorem states that:
[C]∮P dx + Q dy = [R]∬dQ/dx - dP/dy dA

We can use this to simplify the problem. First we find the curl:

dQ/dx - dP/dy = -2x

Now we find the region. It is simply the region between √x and x/2. It is bounded by x=0 and x=4. So we can write:

[R]∬-2x dA = [0, 4]∫ [x/2, √x]∫-2x dy dx

Now we integrate:

[0, 4]∫ [x/2, √x]∫-2x dy dx
[0, 4]∫ -2xy|[x/2, √x] dx
[0, 4]∫ -2x√x + x² dx
x³/3 - 4x^(5/2)/5 |[0, 4]
(64/3 - 128/5) - (0 - 0)
-64/15

You may verify this, if you like, by computing the line integral directly, but it's much more complicated than doing it through green's theorem.

Answer 2

WHAT THE-- GOOD THING I DROPPED OUT OF SCHOOL THREE DAYS AGO!