Trapezoidal Rule

Are you someone trying to understand numerical methods? And thereafter understand the topic of Trapezoidal Rule in detail? To solve this problem Vedantu has brought the Trapezoidal Rule – Definition, Method, Rule, Solved Examples, and Important FAQs for you. The above content is designed by Vedantu’s subject matter experts of maths.   

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 at 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.

\[\int_{a}^{b}f\left ( x \right )\] 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:

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

The error here is:  

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

Xi is a number between a and b.

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Trapezoidal Rule Definition

The 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

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

Solution 1) The area of the trapezoid will be:

A = \[\frac{\left (10+4 \right )\left ( 4 \right )}{2}\] = 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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(Image Will Be Uploaded Soon)

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

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

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

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

h = \[\sqrt{16}\] = 4cm

Look for more trapezoidal rule examples with solutions for practice. 

Fun Facts 

1. In Ancient Greek, Trapezoid was known as τραπέζιον ‘trapézion’ which 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. 

3. Do you know why it is called the trapezoidal rule? When the area under the curve is evaluated, then it is divided into small trapezoids instead of rectangles.

4. It is also known as the Trapezium rule, hence you should not get confused if someone refers to it as the trapezium rule.

Conclusion

Trapezium rule forms one of the most important topics in calculus related to integration, more specifically to definite integrals. You will find questions appearing from this section every year in your exams for at least 3 marks. Apart from this, it has a wide range of applications in higher standards and not just in maths but also in other subjects that you will come across, especially in engineering. After studying this topic, you may want to proceed towards Simpson’s rule, as it gives a more accurate approximate value of the integrals.