Step 1

By Green's Theorem, we have

\(\displaystyle\int_{{C}}{P}{\left.{d}{x}\right.}+{Q}{\left.{d}{y}\right.}=\int_{{R}}{\left(\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}\right)}{d}{A}\)

\(\displaystyle{P}{\left({x},{y}\right)}={x}{y}^{{2}}\)

\(\displaystyle{Q}{\left({x},{y}\right)}={4}{x}^{{2}}{y}\)

\(\displaystyle\frac{{\partial{Q}}}{{\partial{x}}}={8}{x}{y}\)

\(\displaystyle\frac{{\partial{P}}}{{\partial{y}}}={2}{x}{y}\)

\(\displaystyle\oint_{{C}}{\left({x}{y}^{{2}}{\left.{d}{x}\right.}+{4}{x}^{{2}}{y}{\left.{d}{y}\right.}\right)}=\int\int_{{R}}{\left(\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}\right)}{d}{A}\)

\(\displaystyle=\int\int_{{R}}{\left({8}{x}{y}-{2}{x}{y}\right)}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)

Step 2

For line equation

\(\displaystyle{\left({y}-{3}\right)}=\frac{{{3}-{0}}}{{{3}-{0}}}{\left({x}-{3}\right)}\Rightarrow{y}={x}\)

\(\displaystyle{\left({y}-{6}\right)}=\frac{{{6}-{0}}}{{{3}-{0}}}{\left({x}-{3}\right)}\Rightarrow{y}={2}{x}\)

and x varies from 0 to 3

from i \(\displaystyle\oint_{{C}}{\left({x}{y}^{{2}}{\left.{d}{x}\right.}+{4}{x}^{{2}}{y}{\left.{d}{y}\right.}\right)}={\int_{{0}}^{{3}}}{\int_{{x}}^{{{2}{x}}}}{6}{x}{y}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{0}}^{{3}}}{\left[{6}{x}{{\mid}_{{x}}^{{{2}{x}}}}{y}{\left.{d}{y}\right.}\right]}{\left.{d}{x}\right.}\)

\(\displaystyle={\int_{{0}}^{{3}}}{6}{x}{{\left[\frac{{y}^{{2}}}{{2}}\right]}_{{x}}^{{{2}{x}}}}{\left.{d}{x}\right.}\)

\(\displaystyle={\int_{{0}}^{{3}}}\frac{{{6}{x}}}{{2}}{\left[{4}{x}^{{2}}-{x}^{{2}}\right]}{\left.{d}{x}\right.}={\int_{{0}}^{{3}}}{9}{x}^{{3}}{\left.{d}{x}\right.}\)

\(\displaystyle={9}{\int_{{0}}^{{3}}}{x}^{{3}}{\left.{d}{x}\right.}={9}{{\left[\frac{{x}^{{4}}}{{4}}\right]}_{{0}}^{{3}}}=\frac{{{9}\cdot{81}}}{{4}}={182.25}\)

By Green's Theorem, we have

\(\displaystyle\int_{{C}}{P}{\left.{d}{x}\right.}+{Q}{\left.{d}{y}\right.}=\int_{{R}}{\left(\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}\right)}{d}{A}\)

\(\displaystyle{P}{\left({x},{y}\right)}={x}{y}^{{2}}\)

\(\displaystyle{Q}{\left({x},{y}\right)}={4}{x}^{{2}}{y}\)

\(\displaystyle\frac{{\partial{Q}}}{{\partial{x}}}={8}{x}{y}\)

\(\displaystyle\frac{{\partial{P}}}{{\partial{y}}}={2}{x}{y}\)

\(\displaystyle\oint_{{C}}{\left({x}{y}^{{2}}{\left.{d}{x}\right.}+{4}{x}^{{2}}{y}{\left.{d}{y}\right.}\right)}=\int\int_{{R}}{\left(\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}\right)}{d}{A}\)

\(\displaystyle=\int\int_{{R}}{\left({8}{x}{y}-{2}{x}{y}\right)}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)

Step 2

For line equation

\(\displaystyle{\left({y}-{3}\right)}=\frac{{{3}-{0}}}{{{3}-{0}}}{\left({x}-{3}\right)}\Rightarrow{y}={x}\)

\(\displaystyle{\left({y}-{6}\right)}=\frac{{{6}-{0}}}{{{3}-{0}}}{\left({x}-{3}\right)}\Rightarrow{y}={2}{x}\)

and x varies from 0 to 3

from i \(\displaystyle\oint_{{C}}{\left({x}{y}^{{2}}{\left.{d}{x}\right.}+{4}{x}^{{2}}{y}{\left.{d}{y}\right.}\right)}={\int_{{0}}^{{3}}}{\int_{{x}}^{{{2}{x}}}}{6}{x}{y}{\left.{d}{x}\right.}{\left.{d}{y}\right.}\)

\(\displaystyle={\int_{{0}}^{{3}}}{\left[{6}{x}{{\mid}_{{x}}^{{{2}{x}}}}{y}{\left.{d}{y}\right.}\right]}{\left.{d}{x}\right.}\)

\(\displaystyle={\int_{{0}}^{{3}}}{6}{x}{{\left[\frac{{y}^{{2}}}{{2}}\right]}_{{x}}^{{{2}{x}}}}{\left.{d}{x}\right.}\)

\(\displaystyle={\int_{{0}}^{{3}}}\frac{{{6}{x}}}{{2}}{\left[{4}{x}^{{2}}-{x}^{{2}}\right]}{\left.{d}{x}\right.}={\int_{{0}}^{{3}}}{9}{x}^{{3}}{\left.{d}{x}\right.}\)

\(\displaystyle={9}{\int_{{0}}^{{3}}}{x}^{{3}}{\left.{d}{x}\right.}={9}{{\left[\frac{{x}^{{4}}}{{4}}\right]}_{{0}}^{{3}}}=\frac{{{9}\cdot{81}}}{{4}}={182.25}\)