Question

Apply green’s theorem to evaluate the integral 3ydx 2xdy

Answer 1

The value of the integral 3ydx+2xdy using Green's theorem be - xy

The value of    $\int\limits_c 3ydx+2xdy$  be -xy

What is Green's theorem?

Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C.

If M and N are functions of (x, y) defined on an open region containing D and having continuous partial derivatives there, then

$\int\limits_c Mdx+Ndy$ = $\int\int\〖(N_{x}-M_{y}) \;dxdy$

Using green's theorem, we have

$\int\limits_c Mdx+Ndy$ = $\int\int\〖(N_{x}-M_{y}) \;dxdy$ ............................... (1)

Here $N_{x}$ = differentiation of function N w.r.t. x

          $M_{y}$= differentiation of function M w.r.t. y

Given function is: 3ydx + 2xdy

On comparing with equation (1), we get

M = 3y, N = 2x

Now, $N_{x}$ = $\Luge\frac{dN}{dx}$

               = $\frac{d}{dx} (2x)$

              = 2

and, $M_{y}$ = $\Huge\frac{dM}{dy}$

             = $\frac{d}{dy} (3y)$

             = 3

Now using Green's theorem

= $\int\int\〖(2 -3) dx dy$

= $\int\int\ -dxdy$

= $-\int\ x dy$

=$-xy$

The value of  $\int\limits_c 3ydx+2xdy$  be -xy.

Learn more about Green's theorem here:

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