Question

Write the trapezoidal rule formula in numerical methods.

Answer 1

Hint: Mathematics includes the study of topics which are related to quantity, structure, space and change. Trapezoidal method is one of the methods for obtaining an approximate value of definite integral of a curve. So, by using this definition, we can obtain the expression for trapezoidal rule.

Complete step by step answer:
Mathematics is related to all the phenomena occurring in the world. When mathematical structures are good models of real phenomena mathematical reasoning can be used to provide insight or predictions about nature. In mathematics, numerical analysis provides various methods for approximating the value of complex integrals. One of these methods include the method of trapezoidal rule. In this method we can evaluate the area for a given curve by using various strips of equal width which is in the form of trapezoids. The sum of all these trapezoids will give us the area from one point to another or the definite integral of the curve specified.

So, in this way we divide each part into strips of equal width h. Now, we have to find the area of each strip and then sum all the areas to obtain the expression. For an individual strip:

So, the area is divided into triangles and rectangles. And with each succession the area of the previous one is repeated except for the first and the last term strip. So, the final expression for trapezoidal rule is:
$\begin{align}
  & {{I}_{n}}=\dfrac{h}{2}\left\{ {{y}_{0}}+2{{y}_{1}}+2{{y}_{2}}+2{{y}_{3}}+........2{{y}_{n-1}}+{{y}_{n}} \right\} \\
 & {{I}_{n}}=\dfrac{h}{2}\left\{ {{y}_{0}}+{{y}_{n}}+2\sum\limits_{i=1}^{n-1}{{{y}_{i}}} \right\} \\
\end{align}$
In the above expression, ${{y}_{n}}=f({{x}_{n}})\text{ and }{{x}_{n}}={{x}_{n-1}}+h$.

Note: The key concept involved in solving the problem is the knowledge of trapezoidal rule in numerical method analysis. This rule is very important for approximating the definite integral of complex curves. It is very useful in higher studies.