91Ó°ÊÓ

The given information is ${∫}_C(y{e}^x-1)dx+e^{x}dy$.

The Green's theorem connects a line integral around a simple closed curve C to a double integral over the plane region D circumscribed by C in vector calculus. Stokes' theorem has a two-dimensional special case.

Consider an equation and name it as equation 1

${∫}_C(y{e}^x-1)dx+e^{x}dy.......\left(1\right)$

Compare equation (1) to Green’s Theorem.

$$\begin{gathered} P=y{e}^x-1 \\ Q=e^x \end{gathered}$$

Find the differentiation with respect to.

$\frac{∂Q}{∂x}-\frac{∂P}{∂y}=0$

This implies that the surface integral will also be zero, which further leads to the line integral along C is negative that of line integral moving from

$(0,10)to(0,-10).$

Let the value be as follows:

$$\begin{gathered} x=0 \\ dx=0 \end{gathered}$$

Find the Integral.

$$\begin{gathered} l={∫}_{10}^{-10}dy \\ =-\left(-10-10\right) \\ =20 \end{gathered}$$

Hence, the solution to this problem is 20.