Question

Evaluate each of the following integrals in the easiest way you can.

$\mathbf{\oint }\mathbf{(}\mathbf{2}\mathbf{y}\mathbf{d}\mathbf{x}\mathbf{=}\mathbf{3}\mathbf{x}\mathbf{d}\mathbf{y}\mathbf{)}$around the square bounded by x=3, x=5, y=1 and y=3

Short answer

The solution to this problem is /=-20

Step-by-step solution

Given Information.

The given information is$\oint (2\mathrm{ydx}=3\mathrm{xdy})$

Definition of Green’s Theorem.

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.

Find the solution.

Write the values of P and Q.

P=2y

Q=-3x

Find the difference of their differentiation

$$\begin{gathered} \frac{\partial Q}{\partial X}-\frac{\partial P}{\partial y}=-3-2 \\ =-5 \end{gathered}$$

The integral is in the form mentioned below.

$$\begin{gathered} /=-5{\int }_1^3{\int }_3^{5}dxdy \\ =-5\times 4 \\ =-20 \end{gathered}$$

Hence, the solution to this problem is /=-20.