## Saturday, 7 December 2013

### Areas of Plane Curves

Areas in Cartesian Co - ordinate :-

Suppose we want to determine the area A1 bounded by the curve   y = f(x)   , the x - axis an two fixed ordinates   x = a   and   x = b    . The function   f(x)  , is supposed to be single - valued , finite and continuous in the interval  (a , b ) .

The process of finding the area , bounded by any defined contour line is called Quadrature , "the term meaning the investigation of the size of a square which shall have the same area as that  of the region under consideration" .

Here are the example of some Plain Curves :-

Name Implicit equation Parametric equation As a function graph
Straight line $a x+b y=c$ $(x_0 + \alpha t,y_0+\beta t)$ $y=m x+c$
Circle $x^2+y^2=r^2$ $(r \cos t, r \sin t)$
Parabola $y-x^2=0$ $(t,t^2)$ $y=x^2$
Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a \cos t, b \sin t)$
Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a \cosh t, b \sinh t)$