**Summary**

- Trapezium rule can only be applied to Definite Integrals.
- Area under the curve is divided into equally spaced intervals forming a trapezium.
- Trapezium Rule only provides an estimate area of under the curve.
- The formula given is:

where

#### Definite Integrals

Let’s start by briefly recalling the definite integrals from the article ”Integration”.

Definite Integration is when we integrate a function between defined limits. The function must have an upper limit and a lower limit.

It is defined as:

Where *f’(x)* is the derived function of* f* throughout the interval *(a, b)*.

#### Trapezium rule

**Trapezium rule** is also one of the methods to solve a definite integral. To be more precise, when it is difficult or impossible to find the exact value of a given definite integral, we use a method known as “The trapezium Rule” to find its approximate value.

The formula given is:

Where and .

Now suppose, we are to evaluate .

We draw the curve *y = f(x)* between *x = a* and *x = b* and estimate the area under the curve by using various trapeziums.

Suppose we divide the area under the curve between *x = a* to *x = b* into *n* equal strips by taking *(n – 1)* equally spaced coordinates. We then first find the area of each trapezium and then find the sum of areas of all those trapeziums.

So:

Area under the curve =

Where:

Each area between two strips is considered a trapezium and we know that area of a trapezium is given by:

*Note: Increasing number of strips, improves the accuracy of the approximation.*

Hence:

Factorising:

#### Example #1

Q. Estimate the value of using trapezium rule with 6 intervals.

*Solution:*

Width of each interval =

Next we find the value of for all the values of *x* between the intervals.

x | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 |

0 | 0.1105 | 0.2443 | 0.4049 | 0.5967 | 0.8243 | 1.0932 |

Finally we substitute the above calculated values in the trapezium rule formula:

**Ans: **

#### Example#2

Q. Use the trapezium rule with 3 intervals to estimate the value of

*Solution:*

x | 0 | |||

1 | 1.0694 | 1.2909 | 1.7320 |

Finally we substitute the above calculated values in the trapezium rule formula:

**Ans: **