# Use Green's Theorem to evaluate oint_C(x^2+y)dx-(3x+y^3)dy Where c is the ellipse x^2+4y^2=4

Question
Use Green's Theorem to evaluate
$$\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}+{y}^{{3}}\right)}{\left.{d}{y}\right.}$$
Where c is the ellipse $$\displaystyle{x}^{{2}}+{4}{y}^{{2}}={4}$$

2021-02-22
Step 1
It is required to calculate the value of the integral using green’s theorem:
$$\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}-{y}^{{3}}\right)}{\left.{d}{y}\right.}{w}{h}{e}{r}{e}{c}{i}{s}{e}{l}{l}{i}{p}{s}{e}{x}^{{2}}+{4}{y}^{{2}}={4}$$
Step 2
$$\displaystyle\oint_{{C}}{P}{\left.{d}{x}\right.}+{Q}{\left.{d}{y}\right.}=\int\int_{{R}}{\left(\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}\right)}{d}{A}$$
Step 3
So,
$$\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}+{y}^{{3}}\right)}{\left.{d}{y}\right.}=\int\int_{{R}}\frac{\partial}{{\partial{x}}}{\left(-{\left({3}{x}+{y}^{{3}}\right)}\right)}-\frac{\partial}{{\partial{y}}}{\left({x}^{{2}}+{y}\right)}$$
$$\displaystyle=\int\int_{{R}}{\left(-{3}-{1}\right)}{d}{A}$$
$$\displaystyle=\int\int_{{R}}{\left(-{4}\right)}{d}{A}$$
Step 4
$$\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}+{y}^{{3}}\right)}{\left.{d}{y}\right.}=\int\int_{{R}}{\left(-{4}\right)}{d}{A}={\int_{{-{{1}}}}^{{1}}}{\int_{{-{{2}}}}^{{2}}}{\left(-{4}\right)}{\left.{d}{x}\right.}{\left.{d}{y}\right.}$$
$$\displaystyle={\left(-{4}\right)}{\int_{{-{{1}}}}^{{1}}}{{\left[{x}\right]}_{{-{{2}}}}^{{2}}}{\left.{d}{y}\right.}$$
$$\displaystyle={\left(-{4}\right)}{\int_{{-{{1}}}}^{{1}}}{\left({2}-{\left(-{2}\right)}\right)}{\left.{d}{y}\right.}$$
$$\displaystyle={\left(-{4}\right)}{\int_{{-{{1}}}}^{{1}}}{4}{\left.{d}{y}\right.}$$
$$\displaystyle={\left(-{16}\right)}{{\left({y}\right)}_{{-{{1}}}}^{{1}}}$$
=-16(1-(-1))
=-16(2)
$$\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}^{+}{y}^{{3}}\right)}{\left.{d}{y}\right.}=-{32}$$

### Relevant Questions

Evaluate the line integral $$\displaystyle\oint_{{C}}{x}{y}{\left.{d}{x}\right.}+{x}^{{2}}{\left.{d}{y}\right.}$$, where C is the path going counterclockwise around the boundary of the rectangle with corners (0,0),(2,0),(2,3), and (0,3). You can evaluate directly or use Green's theorem.
Write the integral(s), but do not evaluate.
Use Green's Theorem to evaluate $$\displaystyle\int_{{C}}{\left({e}^{{x}}+{y}^{{2}}\right)}{\left.{d}{x}\right.}+{\left({e}^{{y}}+{x}^{{2}}\right)}{\left.{d}{y}\right.}$$ where C is the boundary of the region(traversed counterclockwise) in the first quadrant bounded by $$\displaystyle{y}={x}^{{2}}{\quad\text{and}\quad}{y}={4}$$.
Use Green's Theorem to evaluate the line integral
$$\displaystyle\int_{{C}}{\left({y}+{e}^{{x}}\right)}{\left.{d}{x}\right.}+{\left({6}{x}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}$$
where C is triangle with vertices (0,0),(0,2)and(2,2) oriented counterclockwise.
a)6
b)10
c)14
d)4
e)8
f)12
Use Stokes' theorem to evaluate the line integral $$\displaystyle\oint_{{C}}{F}\cdot{d}{r}$$ where A = -yi + xj and C is the boundary of the ellipse $$\displaystyle\frac{{x}^{{2}}}{{a}^{{2}}}+\frac{{y}^{{2}}}{{b}^{{2}}}={1},{z}={0}$$.
Evaluate $$\displaystyle\int_{{C}}{x}^{{2}}{y}^{{2}}{\left.{d}{x}\right.}+{4}{x}{y}^{{3}}{\left.{d}{y}\right.}$$ where C is the triangle with vertices(0,0),(1,3), and (0,3).
(a)Use the Green's Theorem.
(b)Do not use the Green's Theorem.
Use Green's Theorem to evaluate the line integral. Orient the curve counerclockwise.
$$\displaystyle\oint_{{C}}{F}{8}{d}{r}$$, where $$\displaystyle{F}{\left({x},{y}\right)}={\left\langle{x}^{{2}},{x}^{{2}}\right\rangle}$$ and C consists of the arcs $$\displaystyle{y}={x}^{{2}}{\quad\text{and}\quad}{y}={8}{x}{f}{\quad\text{or}\quad}{0}\le{x}\le{8}$$
Use Green's Theorem to evaluate $$\displaystyle\int_{{C}}\vec{{{F}}}\cdot{d}\vec{{{r}}}$$ where $$\displaystyle\vec{{{F}}}{\left({x},{y}\right)}={x}{y}^{{2}}{i}+{\left({1}-{x}{y}^{{3}}\right)}{j}$$ and C is the parallelogram with vertices (-1,2), (-1,-1),(1,1)and(1,4).
$$\displaystyle\int_{{C}}{x}{y}^{{2}}{\left.{d}{x}\right.}+{4}{x}^{{2}}{y}{\left.{d}{y}\right.}$$
Use Stokes' Theorem to compute $$\displaystyle\oint_{{C}}\frac{{1}}{{2}}{z}^{{2}}{\left.{d}{x}\right.}+{\left({x}{y}\right)}{\left.{d}{y}\right.}+{2020}{\left.{d}{z}\right.}$$, where C is the triangle with vertices at(1,0,0),(0,2,0), and (0,0,2) traversed in the order.
Find the work done by the force field F(x,y)=4yi+2xj in moving a particle along a circle $$\displaystyle{x}^{{2}}+{y}^{{2}}={1}$$ from(0,1)to(1,0).