Question

# Use Green's Theorem to evaluate the line integral int_C(y+e^x)dx+(6x+cosy)dy 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 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

2021-03-13

Step 1
Given $$\displaystyle{P}={y}+{e}^{{x}}{\quad\text{and}\quad}{Q}={6}{x}+{\cos{{y}}}$$
Green's theorem is
$$\displaystyle\int_{{C}}{P}{\left.{d}{x}\right.}+{Q}{\left.{d}{y}\right.}=\int\int_{{C}}{\left(\frac{{\partial{Q}}}{{\partial{x}}}-\frac{{\partial{P}}}{{\partial{y}}}\right)}{\left.{d}{x}\right.}{\left.{d}{y}\right.}$$
$$\displaystyle=\int\int_{{C}}{\left({6}-{1}\right)}{\left.{d}{x}\right.}{\left.{d}{y}\right.}$$
$$\displaystyle={5}\int\int_{{C}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}$$
Since C is closed triangle with vertices (0,0) , (0,2) and (2,2) which is oriented counter clockwise .
Since $$\displaystyle\int\int_{{C}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}$$ is the area of that triangle.
Step 2
Then
$$\displaystyle\int\int_{{C}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}=\frac{{1}}{{2}}\cdot{2}\cdot{2}={2}$$
Hence
$$\displaystyle\int_{{C}}{\left({y}+{e}^{{x}}\right)}{\left.{d}{x}\right.}+{\left({6}{x}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}={5}\cdot\int\int_{{C}}{\left.{d}{x}\right.}{\left.{d}{y}\right.}$$
$$=5\times2=10$$
Therefore (2) option is correct.