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
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	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$(a \cosh t, b \sinh t)$

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Areas of Plane Curves

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