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LARCALC$1215\mathrm{.}4\mathrm{.}008.$

Verify Green's Theorem by evaluating both integrals

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}{y}^{2}dx+x^{2}dy={\displaystyle \underset{R}{\overset{\phantom{}}{}}}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(\frac{\text{d}\text{e}\text{l}\text{N}}{\text{d}\text{e}\text{l}\text{x}}-\frac{\text{d}\text{e}\text{l}\text{M}}{\text{d}\text{e}\text{l}\text{y}})dA$

for the given path.

$C$ : rectangle with vertices $(0,0),(2,0),(2,5),$ and $(0,5)$

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}{y}^{2}dx+x^{2}dy=$

${\displaystyle \underset{R}{\overset{\phantom{}}{}}}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(\frac{\text{d}\text{e}\text{l}\text{N}}{\text{d}\text{e}\text{l}\text{x}}-\frac{\text{d}\text{e}\text{l}\text{M}}{\text{d}\text{e}\text{l}\text{y}})dA=$

$$