Use Green's Theorem to evaluate the line integral. Orient curve counterclockwise unless otherwise indicated .C(lnx+y)dx−x2dy where C is the rectangle with vertices (1, 1), (3, 1), (1, 4), and (3, 4).

Answer 1

Green’s theorem finds the line integral around of a space by finding the area integral of the area enclosed by the line:

∫C Pdx + Qdy = ∫∫D(∂Q/∂x - ∂P/∂y)dxdy

In this case, 

P = ln(x) + y 

Q = -x^2

The partial derivatives can be found:

∂Q/∂x = -2x

∂P/∂y = 1

By plotting the points, it's apparent that x varies from 1 to 3, and y varies from 1 to 4 over the area. These are used as the limits of integration for the double integral:

∫(1,4) ∫(1,3) -2x-1 dxdy

=∫(1,4) [-x^2-x](1,3) dy

=∫(1,4) (-10) dy

=[-10y] (1,4)

=-30