## Trapezoidal Rule

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### Description

In numerical analysis, this rule is a technique for approximating the definite integral. In this rule, the approximation is done on the region under the graph of f(x) as a trapezoid and then the area of that trapezoid is calculated. The result of the trapezoidal rule tends to be more accurate than the other methods.

DERIVATION

A mechanical quadrature formula will be called closed or open according to the limits of integration are used as interpolating points or not.

Sometimes it is easy to calculate the integration by breaking the whole interval into some subintervals. After computing each subinterval we have to add it to get the actual result.

By using newton cote’s formula we get,

Then,

Hence,

Now, if we apply newton cote’s formula with n = 1, the trapezoidal formula can be deduced.

Hence,

GRAPHICAL REPRESENTATION

### Algorithm

INPUT:

A function f(x)

OUTPUT:

The value after performing the integration

PROCESS:

``````Step 1: [defining a function as a macro F(x)]
F(x) 4x-3x^2

Step 2: [Trapezoidal Method]
Set a ← 0.0 and b ← 1.0 [the lower limit and upper limit of the integration]
Read n [the number of intervals]
Set h ← (b - a)/n
Set m ← F(a)
Set t ← F(b)
Print m
Print t
for i = 0 to n - 2 repeat
Set z ← z + h
Set y ← F(z)
Set sum ← sum + y
Print z, y
[End of ‘for’ loop]
Set sum ← 2 × sum
Set s ← (h/2) × (sum + m + t)
Print s
[End of ‘trapezoidal’ method]``````

### Code

1. The trapezoidal rule is more accurate.

2. This rule is simple and ideal for many integration tasks.