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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 Gerade.svg
Circle x^2+y^2=r^2 (r \cos t, r \sin t)
framless
Parabola y-x^2=0 (t,t^2) y=x^2 Parabola.svg
Ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a \cos t, b \sin t)
framless
Hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a \cosh t, b \sinh t)







Hyperbola.svg