Textbook Problem

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Verifying Green’s TheoremIn Exercises 9 and 10, verify Green’s Theorem by using a computer algebra system to evaluate both integrals ${\displaystyle {\int }_{c}x{e}^{y}dx+e^{x}dy}={\displaystyle {\int }_R{\displaystyle \int \left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)}dA}$ for the given path.

C: boundary of the region lying between the graphs of $y=x$ and $y=x^3$ in the first quadrant

To determine

By the use of a computer algebra system the value of both the integrals ${\displaystyle \underset{C}{\int }x{e}^{y}dx+e^{x}dy={\displaystyle \underset{R}{\iint }(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y})dA}}$ for the given path C and also, verify Green’s Theorem.

Explanation of Solution

Given:

C: boundary of the region lying between the graphs of $y=x$ and $y=x^3$ in the first quadrant.

Formula used:

$\frac{\partial }{\partial x}{e}^x=e^x$

$\frac{\partial }{\partial y}{e}^y=e^y$

Calculation:

The path C can be considered that is composed of two paths C ₁ and C ₂. The path C ₁ given by $y=x$ and C ₂ given by $y=x^3$, which intersects each other at $(0,0)$ & $(1,1)$. The parametric equation of the two paths gives us

$r(t)=\{\begin{array}{cc}ti+t^{3}j;& 0\le t\le 1\\ (2-t)i+(2-t)j;& 1\le t\le 2\end{array}$

For the path C ₁, has $x(t)=t$ and $y(t)=t^3$, which implies that $dx=dt$ and $dy=3t^{2}dt$

For the path C ₂, has $x(t)=2-t$ and $y(t)=2-t$, which implies that $dx=-dt$ and $dy=-dt$

$\begin{array}{c}{\displaystyle \underset{C}{\int }x{e}^{y}dx+e^{x}dy}={\displaystyle \underset{C_1}{\int }\left(x{e}^{y}dx+e^{x}dy\right)}+{\displaystyle \underset{C_2}{\int }\left(x{e}^{y}dx+e^{x}dy\right)}\\ ={\displaystyle \underset{0}{\overset{1}{\int }}\left(t{e}^{t^3}dt+e^{t}3t^{2}dt\right)}+{\displaystyle \underset{1}{\overset{2}{\int }}\left((2-t)e^{2-t}(-dt)+e^{2-t}(-dt)\right)}\\ ={\displaystyle \underset{0}{\overset{1}{\int }}\left(t{e}^{t^3}+e^{t}3t^2\right)dt}+{\displaystyle \underset{1}{\overset{2}{\int }}{e}^{2-t}\left(t-3\right)}dt\\ =2.935997-2.718300\\ =0.217697\end{array}$

The value of the two integrals ${\displaystyle \underset{0}{\overset{1}{\int }}\left(t{e}^{t^3}+e^{t}3t^2\right)dt}$ and ${\displaystyle \underset{1}{\overset{2}{\int }}{e}^{2-t}\left(t-3\right)}dt$ can be by the use of a computer algebraic system