Question

Use green’s theorem to evaluate

Answer 1

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}`