Multivariable Calc. problem. How would you use the Green's Theorem to evaluate the line integral below...?

Question

Using the Green's Theorem to evaluate the line integral

∫ P(x, y) dx+ Q(x, y) dy
C

where C is the path from (0, 0) to (1, 0) along y = 0; then from (1, 0) to (1, 1) along x = 1; then from (1, 1) to (0, 0) along y = x^1/2, given that P(x, y) = y^8, and Q(x, y) = 6xy.

This are the possible answers in the back of the book....

a. 1
b. 11/10
c. 7/6
d. 17/14
e. 5/4
f. 23/18
g. 13/10
h. 29/22
i. 4/3

or is it none of these?

mathematician · Accepted Answer

Well, dQ/dx=6y and dP/dy=8y^7, so Green's theorem says that the integral is the same as the double integral over the region contained of 
6y-8y^7
The limits of the double integral will be 0 to 1 for x and 0 to x^1/2 for y. The inside integral will give 3y^2-y^8 evaluated from 0 to x^1/2, so
3x-x^4.
This is then integrated from 0 to 1 to give (3/2)x^2-(1/5)x^5 from 0 to 1, or
3/2-1/5=13/10.

сhееsеr1 · Answer

Well, let's take a look at the theorem:
http://en.wikipedia.org/wiki/Greene%27s_theorem

This means we want to compute:

â Q / âx = 6y
â P / ây = 8y^7

â Q / âx - â P / ây  = 6x

So we're simply looking at:

â«â« (6y - 8y^7) dydx

over the unit square between the origin and (1,1).

We get:
1 1
â«â« (6y - 8y^7) dxdy
0 0

1
â« (6y -  8y^7) dy
0


3y^2 - y^8 from y=0 to y=1

3 - 1 = 2

enomoney · Answer

a.

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Multivariable Calc. problem. How would you use the Green's Theorem to evaluate the line integral below...?

3 Answers