Use green’s theorem to evaluate

Question

asked Nov 2, 2022 163k views

1 Answer

Chama · Answer 1 · 2022-11-06T19:12:18+0000

By Green's theorem (all the conditions are met), we have

$\displaystyle \int_C \sqrt y \, dx + \sqrt x \, dy = \iint_D (\partial(\sqrt x))/(\partial x) - (\partial(\sqrt y))/(\partial y) \, dx \, dy$

where D is the interior of the path C, or the set

$D = \left\{ (x, y) : 0 \le y \le \frac{x^2}2 \text{ and } 0 \le x \le 2 \right\}$

So, the line integral reduces to the double integral,

$\displaystyle \frac12 \int_0^2 \int_0^{\frac{x^2}2} x^(-\frac12) - y^(-\frac12) \, dy \, dx$

$\displaystyle = \frac12 \int_0^2 x^(-\frac12)\left(\frac{x^2}2\right) - 2\left(\frac{x^2}2\right)^(\frac12) \, dx$

$\displaystyle = \frac12 \int_0^2 \frac12 x^(\frac32) - \sqrt 2 \, x \, dx$

$\displaystyle = \frac14 \int_0^2 x^(\frac32) - 2\sqrt 2 \, x \, dx$

$\displaystyle = \frac14 \left(\frac25\cdot2^(\frac52) - \sqrt2\cdot2^2\right) = \boxed{-\frac{3\sqrt2}5}$