Trapezoidal Rule

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Integration is a very important concept in calculus and the main purpose of finding integrals of various functions is to find the area of the region covered under the graph of the function. Trapezoidal or Trapezium rule is a method in which we can calculate the area under the graph of a given function. In other words, we are evaluating the integral of the given function since finding area implies finding integral! In Trapezoidal rule, the region under the graph is divided into several trapeziums and the area is estimated.
 
Example 1: Evaluate∫ √(2x) dx where ‘x’ is going from 0 to 1.

Here the values of ‘x’ are divided into 4 equal intervals as 0, 0.25, 0.50, 0.75 and 1.

These numbers are regularly spaced, and hence the space, h = 0.25.

If x= 0, then √(2x) = 0

If x= 0.25, then √(2x) = 0.707

If x= 0.50, then √(2x) = 1

If x= 0.75, then √(2x) = 1.22

If x= 1, then √(2x) = 1.414
 
Applying the trapezoid rule we get: 1/2* 0.25* (0+ 1.414+ 2(0.707+ 1+ 1.22))

This gives the area= 0.9085
 
Example 2: Evaluate ∫ √(3x+ 1) dx where ‘x’ is going from 0 to 1.

Here the values of ‘x’ are divided into 4 equal intervals as 0, 0.25, 0.50, 0.75 and 1.

These numbers are regularly spaced, and hence the space, h = 0.25.

If x= 0, then √(3x+ 1) = 1

If x= 0.25, then √(3x+ 1) = 1.75

If x= 0.50, then √(3x+ 1) = 2.50

If x= 0.75, then √(3x+ 1) = 3.25

If x= 1, then √(3x+ 1) = 2
 
Applying the trapezoid rule we get: 1/2* 0.25* (1+ 1.75+ 2(1.75+ 2.50+ 3.25))

This gives the area= 2.22
 

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