Question

# Let f = [x^2y^2, -x/y^2] and R : 1 <= x^2 + y^2 ,+ 4, x >= 0, y>= x. Evaluate int_C F(r)* dr counterclockwise around the boundary C of the region R by Green's theorem.

Let $$\displaystyle{f}={\left[{x}^{{2}}{y}^{{2}},-\frac{{x}}{{y}^{{2}}}\right]}$$ and $$\displaystyle{R}:{1}\le{x}^{{2}}+{y}^{{2}},+{4},{x}\ge{0},{y}\ge{x}$$. Evaluate $$\displaystyle\int_{{C}}{F}{\left({r}\right)}\cdot{d}{r}$$ counterclockwise around the boundary C of the region R by Green's theorem.

2020-12-17
Step 1
Given that, $$\displaystyle{F}={\left[{x}^{{2}}{y}^{{2}},-\frac{{x}}{{y}^{{2}}}\right]}$$ and $$\displaystyle{R}:{1}\le{x}^{{2}}+{y}^{{2}}\le{4},{x}\ge{0},{y}\ge{x}$$.
Here,
$$\displaystyle{F}_{{1}}={x}^{\gamma}{y}^{\gamma}{1}\le{r}\le{2}$$
$$\displaystyle{F}_{{2}}=-\frac{{x}}{{y}^{{2}}}\frac{\pi}{{4}}\le{0}\le\frac{\pi}{{2}}$$
$$\displaystyle\frac{{{d}{F}_{{2}}}}{{{\left.{d}{x}\right.}}}=-\frac{{1}}{{y}^{{2}}}$$
$$\displaystyle\frac{{{d}{F}_{{1}}}}{{{\left.{d}{y}\right.}}}={2}{y}{x}^{{2}}$$
Step 2
Then, by Green's theorem,
$$\displaystyle\int_{{C}}{F}{d}{r}=\int\int_{{R}}{\left(\frac{{{d}{F}_{{2}}}}{{{\left.{d}{x}\right.}}}-\frac{{{d}{F}_{{1}}}}{{{\left.{d}{y}\right.}}}\right)}{\left.{d}{x}\right.}{\left.{d}{y}\right.}$$
Consider the polar coordinates.
$$\displaystyle{x}={r}{\cos{{0}}}$$
$$\displaystyle{y}={r}{\sin{{0}}}$$
j = r
That is, dx dy = r dr d0
Step 3
Evaluate the integral as follows.
$$\displaystyle\int_{{C}}{F}{d}{r}={\int_{{\frac{\pi}{{4}}}}^{{\frac{\pi}{{2}}}}}{\int_{{1}}^{{2}}}{\left[-\frac{{1}}{{{r}^{{2}}{{\sin}^{{2}}{0}}}}-{2}{r}^{{3}}{{\cos}^{{2}}{0}}{\sin{{0}}}\right]}{r}{d}{r}{d}{0}$$
$$\displaystyle={\int_{{\frac{\pi}{{4}}}}^{{\frac{\pi}{{2}}}}}{{\left[\frac{{-{\ln{{r}}}}}{{{{\sin}^{{2}}{0}}}}-\frac{{{2}{r}^{{5}}{{\cos}^{{2}}{0}}}}{{5}}{\sin{{0}}}\right]}_{{1}}^{{2}}}{d}{0}$$
$$\displaystyle={\int_{{\frac{\pi}{{4}}}}^{{\frac{\pi}{{2}}}}}{\left(\frac{{-{\ln{{2}}}}}{{{{\sin}^{{2}}{0}}}}-\frac{{{2}{\left({31}\right)}{{\cos}^{{2}}{0}}}}{{5}}\right)}{\sin{{0}}}{)}{d}{0}$$
$$\displaystyle={\ln{{2}}}{\left(\frac{{\cot{\pi}}}{{4}}-\frac{{\cot{\pi}}}{{4}}\right)}+\frac{{62}}{{5.3}}{\left({{\cos}^{{3}}{\left(\frac{\pi}{{2}}\right)}}-{{\cos}^{{4}}{\left(\frac{\pi}{{4}}\right)}}\right)}$$
$$\displaystyle=-{\ln{{2}}}=\frac{{31}}{{{15}\sqrt{{{2}}}}}$$
Therefore, $$\displaystyle\int_{{C}}{F}{d}{r}=-{\ln{{2}}}-\frac{{31}}{{{15}\sqrt{{{2}}}}}$$