Trapezoidal Rule

Trapezoidal Method

A numerical method is a tool that makes use of algebraic and arithmetic techniques, to solve systems of complex equations in an approximate way. There are equations that we find to be extremely difficult to be solved analytically, in such a case, if we reformulate these with arithmetic operations, it is possible to give an approximate solution to them. There are numerous applications of numerical methods, such as solving integrals, or derivatives, and even approximations. In integrals, to give an approximate solution to several algorithms, we can make an implementation of the trapezoidal rule or trapezoidal method in scale. In Mathematics, the trapezoid rule or trapezoidal method is basically a numerical integration method. In other words, it is a method with the help of which we can calculate approximately the value of the definite integral.


Trapezoidal Rule

As stated above, the trapezoid rule in mathematics is a numerical integration method that we use to calculate the approximate value of the definite integral.

                                                 \[\int_{a}^{b}f(x)dx\]                                     

The rule is on the basis of an approximating value of the integral of f (x) by that of the linear function that passes through the points (a, f (a)) and (b, f (b)). The integral of this is always equal to the area of the trapezoid under the graph of the linear function. It follows that:

                               \[\int_{a}^{b}f(x)dx\]     = \[(b-a)\frac{f(a)+f(b)}{2}\]

The error here is:

                                       \[-\frac{(b-a)^3}{12}f^{(2)}(੬)\]

                              

Xi is a number between a and b.

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Trapezoidal Rule Definition

Trapezoidal Rule is mostly used for evaluating the area under the curves. This is possible if we divide the total area into smaller trapezoids instead of using rectangles. The Trapezoidal Rule integration actually calculates the area by approximating the area under the graph of a function as a trapezoid. Then the average of the left and the right sum is taken. The accurate value of the Trapezoidal may not be so true as compared to Simpson’s Rule when the underlying function is smooth. The reason behind this is that Simpson’s Rule makes use of the quadratic approximation instead of linear approximation. Simpson’s Rule as well as Trapezoidal Rule give the approximation value, but the result of Simpson’s Rule has an even more accurate approximation value of the integrals.

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Trapezoidal Rule Examples With Solutions

Example 1) Here are a few trapezoidal rule examples with solutions. Find the area of the following trapezoid:

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Solution 1) The area of the trapezoid will be:

                   A = \[\frac{(10+4).4}{2}\] = \[28cm^2\]

Question 2) Given below is a figure of a square ABCD where a line joins the midpoint of BC (Point E) with Point D. If the square has a side of 4 m, calculate the area of the acquired trapezoid shape.

Solution 2) If the square has a side of 4 m, we can calculate the area of the acquired trapezoid

shape as: 

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                                      A = \[\frac{(4+2).4}{2}\] = \[12cm^2\]

Question 3) Given below is an isosceles trapezoid, find the height. 

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Solution 3) Find the height of the following isosceles trapezoid:

                               \[5^2\] = \[h^2 + 3^2\]

                             h = √16 = 4cm

Look for more trapezoidal rule examples with solutions for practice.  


Fun Facts 

  1. In Ancient Greek, Trapezoid was known as τραπέζιον ‘trapézion’ which literally meant “a little table” and which is why it also refers to “irregular quadrilateral”. 

  2. The term trapezium was introduced in the English language in 1570 by Marinus Proclus. He was the first person who used the word trapezoid in the first book of Euclid’s Elements for the first time. 

FAQ (Frequently Asked Questions)

Question 1: When do we use Simpson's r 1/3d rule and the Trapezoidal Rule?

Answer: The use of Simpson’s 1/3rd rule and the trapezoidal rule are stated below: 

Simpson’s 1/3 Rule: By now know that the trapezoidal rule is actually based on approximating the integrand by a first-order polynomial followed by integrating the polynomial over an interval of integration. Simpson’s 1/3 rule can be known for an extension of the Trapezoidal rule where the integrand is approximated not by the first-order polynomial but by a second-order polynomial.

Trapezium Rule: The trapezium rule is a method with which we can find the estimated value of the value of an integral between specified limits. We can also express it in other words as the trapezium rule is a system of calculating the area under a curve on a graph.

Question 2: Why do we use trapezoidal and Simpson’s rule in evaluating definite integrals?

Answer: The answer to the above-mentioned question is because these rules help us in approximating the values of definite integrals fairly accurately without locating a primitive (i.e. the indefinite integral) which at times turns out to be difficult​ and sometimes even impossible. One just requires the values of the function at certain equispaced points in the interval of integration. Especially Simpson's (1/3)- and (3/8)-rules that astonishingly results in accurate values without much effort. In most of the suitable situations, one just needs the values of the specific integrals rather than the formulas of the indefinite integrals. In several situations, we don't have a formula for the integrand but only hold a few data points that give the values of the integrand, and in these cases, the numerical methods of integration prove not only convenient but also indispensable.