Question

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

By Green's theorem,

`\displaystyle \int_C P(x,y)\,dx + Q(x,y)\,dy = \iint_D (\partial Q)/(\partial x) - (\partial P)/(\partial y) \, dA`

where
`D` is the interior of
`C`. It's easy to see that

`D = \left\{(x,y) ~:~ 1 \le x \le 3 \text{ and } 1 \le y \le 4\right\}`

Now,

`(\partial Q)/(\partial x) = (\partial)/(\partial x)(-x^2) = -2x`

`(\partial P)/(\partial y) = (\partial)/(\partial y)(\ln(x) + y) = 1`

so that the line integral reduces to

`\displaystyle \int_C (\ln(x)+y)\,dx - x^2\,dy = -\int_1^3 \int_1^4 (2x+1) \, dy\,dx = \boxed{-30}`