1. Mid Point Method
As we know that area = 
2. So, if we know the area of the circle and it's
radius,
we can calculate the value of .
In this method, we take a quarter of the circle and divide it in different equal parts and we can calculate
area by integration of all the parts. So, if we are getting 1/4 th area then multiplying by 4r2, we can get the value of .
In this method if division of the circle is more, approximate area is closer and then 's
value is more approximate. In this applet, we are dividing the area using different methods. In this Applet division
of circle is in multiple of two for calculating convention.
The Applet is divided in 1, 2, 4, ........................., 256 parts. All the methods are
as following :
2. Left End Point Method
3. Right End point Method
4. Simpson Method
5. Trapezoidal Method
Now, we shall discuss all the above methods in detail.
1. MID POINT METHOD :
In this method, we divide the circle in the no. of parts selected and height of the rectangle is
considered as the mid point
of the both end of the rectangle. This method gives better approximation than other methods.
2. LEFT END POINT METHOD :
In Left end point method, the height of the rectangle is calculated at the left point of the rectangle.
It gives less approximation than Mid point method but gives better approximation than Right end point method.
3. RIGHT END POINT METHOD :
In Right end point method, the height of the rectangle is calculated as the right end point of the circle. This methods
gives less approximate method.
4. SIMPSON METHOD :
This method is given by Simpson. In this method, we have to draw arc instead of rectangle, which require at least three
points. So after dividing the circle we will draw the arc and calculate the area using Simpson method.
Simpson's Rule for calculating area is given as :
f(x)dx = h/3 [y0 + 4y1 + 2y2 + 4y3 + 2y4 +...........+ 2yn-2 + 4yn-1 + yn]
where h = (a - b) / n.
5. TRAPEZOIDAL METHOD :
In this Trapezoidal method, we join the top of the rectangle by straight lines instead of curves and calculate the area
using Trapezoidal method. Trapezoidal method is given below :
f(x)dx = h/2 [y0 + 2y1 + 2y2 + 2y3 + 2y4 + ........... + 2yn-1 + yn]
where h = (b - a) / 2n.
HOW TO USE THIS APPLET :
In this method, first of all we shall select one of the five radio buttons,
which identify the method though which we have to calculate the value of .
The selected radio button will be red and rest of the buttons will be green.
If at any moment of time we select different radio button, then
selected radio button will be red and others will be green.
Now, we will select the no. of parts in which we have to divide the circle.
It will be in the form of 1, 2, 4, …, 256 according to programming convention.
Means it is in multiple of two and then its area is calculated internally and
gives the value of in the text box. The value of
will differ in different methods for the number of same parts.
