Numerical Integration- Trapezoidal Rule: Formula and Application

Numerical Integration is an important part of mathematics and the Trapezoidal rule formula helps in deriving this integration through the use of graphs. It is based on the graphical representation of a function, say f(x). The function, when plotted on the graph, takes the shape of a trapezoid, and then under the Trapezoidal rule, we calculate that the graphical area of that trapezoid is so formed.

Earlier, mathematicians visualised the shape of the integral function as a rectangle but they realised that it is not giving accurate results. Then, the trapezoidal rule was formulated to bring accuracy. 

Under the Composite trapezoidal rule, we use a simple formula to find the approximate graphical area rather than drawing the graph.

What is the Trapezoidal rule formula: The meaning and exact formula

Trapezoidal rules are derived from the Newton-Cotes formula. The basic idea behind this formula is to sub-divide the intervals and get the right value of integral.

To use this formula, you have to first divide and find the nth order polynomial. This means that you will first divide the intervals of the function into several equal subintervals, called ‘n’. This n will be provided to you in the question. Using that, you have to estimate the integral value by putting the number in the given formula.

The interval limits of the integral (a and b) are also provided in the question itself.

The formula used for deriving integral value through Trapezoidal rule is:

∫ba f(x)dx = Delta x/2 [ f(x0)+ 2f(x1)+2f(x2)… 2f(xn-1)+f(xn)]

Here, f(x) is the function with a and b are intervals. The value of delta x is obtained by dividing n by (b-a). So, Delta x= (a-b)/n.

Further, Value of xi= a+i(Delta x) i.e.

if i=1, x1= a+Delta x

if i=2, x²= a+ 2(Delta x), and so on.

Once you have got all the values, you can put them in a formula and get the right answer.

Why is the name of the formula so?

One question that comes to mind when we read about the trapezoidal rule is why name it on trapezoid and why not any other shape. So the answer is simple. The name of any formula is mostly based on the mathematical concepts that are used in that formula or the scientist who had discovered this formula.

In the case of the trapezoidal rule, the name is so because the whole area is divided into small trapeziums for calculating the graphical area of the function. Earlier rectangles were used for calculating the area between given intervals. But now this formula devised a new approach based on trapezoids for the area covered by a curve. This is why this formula is called the trapezoidal rule.

Trapezoidal rule examples:

To get more clarity on the application of the Trapezoidal rule, let’s check out some Trapezoidal rule examples.

Ques: If n=4, what would be the value of integral ∫51 f(x)dx if f(x)= square root of (1+x2)

Ans: Here, delta x= (b-a)/n

I.e. Delta x= {5-1}/4

So,  Delta x = 1

This means that we will keep an interval of one and we need to find values of x0, x1, x2, and x3.

Sub intervals will be a=1, 2, 3, 4, and b=5.

X0= a+0(delta x) {refer to the formula section where formula for xi is given}

So x0= 1,

Similarly, x1= 1+1(delta x)= 2, x2= 3, x3=4, x4= 5

Next, we will calculate the function of x1, x2, x3, and x4.

For calculating, f(x0)= square root of x0 =1

F(x1)= (√2 = 1.41)

F(x2) =square root of (10) {because x2=3 and when put in f(x) it equals to square root of 1+32= 10}

F(x3) =square root of 17

F(x4) =square root of 26.

Now we will put all these derived values in the formula ∫ba f(x)dx = Delta x/2 [ f(x0)+ 2f(x1)+2f(x2)… 2f(xn-1)+f(xn)]

And you will get the answer. 

Sometimes, you are given a table in which the values of x, f(x), and n are already given and you have to use the Composite trapezoidal rule for calculating the area of the curve formed by that mathematical function. In that case, most of your work is already done. So, you just need to put the values in the formula and find the area.

Conclusion:

The trapezoidal rule was discovered to make it easy for the mathematicians to find the area of graphical representation of a mathematical function. This formula based on Newton-Cotes formula is helpful when rectangle visualisation fails to give the right result for integration. From an exam perspective, whenever you will be asked about Composite trapezoidal rule, you will mostly have the value of n, a, and b using which you have to find the value of the integral given.

For conceptual clarity, you should solve more and more examples on this formula so that you become thorough with the concept.