Question

Use green's theorem to evaluate the line integral along the given positively oriented curve. c yex dx 2ex dy, c is the rectangle with vertices (0, 0), (4, 0), (4, 3), and (0, 3)

Answer 1

The line integral along the given positively oriented curve  is  mathematically given as

$=3\left[e^{4}-1\right]$

This is further explained below.

What is the line integral along the given positively oriented curve.?

Generally,

$\int M d x+N d y=\iint\left(\frac{\partial N}{\partial x}-\frac{\partial M}{2 y}\right) d y d x$

M=y e^{x}

Therefore

x -->0 to 4

y --> 0 to 3

$\end{aligned}\\&=\int_{0}^{4} \int_{0}^{3}\left(2 e^{x}-e^{x}\right) d y d x\\&=\int_{0}^{4} \int_{0}^{3} e^{x} d y d x\\&=3 \int_{0}^{4} e^{x} d x\\&=3\left[e^{x}\right]_{0}^{4}\\&=3\left[e^{4}-e^{0}\right]\\&=3\left[e^{4}-1\right]\end{aligned}$  

In conclusion, the line integral along the given positively oriented curve.

$=3\left[e^{4}-1\right]$

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