Derivation of Trapezoidal Rule Formula
The Trapezoidal Rule formula for calculating the area under the curve is derived by dividing the area under the curve into several trapezoids and then finding their sum.
Statement:
Let f(x) be a continuous function defined on the interval (a, b). Now we divide the intervals (a, b) into n equal sub-intervals where the width of each interval is,
Δx = (b – a)/n
such that a = x0 < x1 < x2 < x3 <…..< xn = b
Then the Trapezoidal Rule formula is,
∫baf(x) dx ≈ △x/2 [f(x0) + 2f(x1) + 2f(x2) +….2f(xn-1) + f(xn)]
where, xi = a + i△x
If n → ∞, the R.H.S of the expression gives the definite integral
Proof:
This formula is proved by dividing the area under the given curve as shown in the above figure into various trapezoids. The first trapezoid has a height Δx and the length of parallel bases are f(x0) and f(x1)
The area of the first trapezoid = (1/2) Δx [f(x0) + f(x1)]
Similarly, the area of the remaining trapezoids are (1/2)Δx [f(x1) + f(x2)], (1/2)Δx [f(x2) + f(x3)], and so on.
Now we can say that,
∫ba f(x) dx ≈ (1/2)Δx (f(x0)+f(x1) ) + (1/2)Δx (f(x1)+f(x2) ) + (1/2)Δx (f(x2)+f(x3) ) + … + (1/2)Δx (f(xn-1) + f(xn) )
After simplifying we get,
∫ba f(x) dx≈ (Δx/2) (f(x0)+2 f(x1)+2 f(x2)+2 f(x3)+ … +2f(xn-1) + f(xn))
Thus, the trapezoidal rule is proved.
Trapezoidal Rule
The trapezoidal rule is one of the fundamental rules of integration which is used to define the basic definition of integration. It is a widely used rule and the Trapezoidal rule is named so because it gives the area under the curve by dividing the curve into small trapezoids instead of rectangles.
Generally, we find the area under the curve by dividing the area into smaller rectangles and then finding the sum of all the rectangles, but in the trapezoidal rule the area under the curve is divided into trapezoids, and then their sum is calculated. The trapezoidal rule is used to find the value of the definite integrals in numerical analysis. This rule is also called the trapezoid rule or the trapezium rule. Let us learn more about the trapezoidal rule, its formula and proof, example, and others in detail in this article.
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