Mumbai University > Mechanical Engineering > Sem 4 > Applied Mathematics IV

Marks: 5M

Year: May 2015

Given:

$$\int{\boldsymbol{xdy}\boldsymbol{-}\boldsymbol{ydx}} -------- (1)$$

Comparing Equation (1) with $\int{Pdx+\ Qdy}$

We get P = -y and Q = x

$$\therefore \frac{\partial P}{\partial y} = -1 , \frac{\partial Q}{\partial x}=1$$

By Green Theorem, $$\int{Pdx+\ Qdy}=\ \int{\int{(\frac{\partial Q}{\partial x}-\ \frac{\partial P}{\partial y}}})\ dx\ dy$$ $$\int{\boldsymbol{xdy}\boldsymbol{-}\boldsymbol{ydx}\boldsymbol{\ }}=\ \int{\int{\left(1\right)-\left(-1\right)dxdy}}$$ $$ = \int{\int_R{2\ dx\ dy}}$$ $$= 2 * Area \ bounded \ by \ curve$$

$\boldsymbol{\therefore }\boldsymbol{\ }\frac{\boldsymbol{1}}{\boldsymbol{2}}$$\int{\boldsymbol{xdy}\boldsymbol{-}\boldsymbol{ydx}}\boldsymbol{=\ }$Area bounded by curve

Hence proved.