Trapezoidal Rule Formula: Definition & Solved Examples

Trapezoidal rule, also referred to as the trapezoid rule or trapezium rule, is a quantitative analytic technique for approximating the definite integral. The trapezoidal rule is an integrating rule that divides a curve into little trapezoids to compute the area beneath it. The surface under the curve is calculated by adding the areas of all the tiny trapezoids. Among the most significant rules in theoretical approach is this rule. Any region that measures the total is subdivided into several sections. 

Read Also: Volume of a Right Circular Cone

Key Terms: Trapezoidal rule, Trapezoid formula, Equation, Integration, approximation, value, integrals, trapezoids, area, definite integral

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Trapezoidal Rule

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Instead of utilising rectangles, the Trapezoidal Rule divides the whole area into tiny trapezoids to find the space under the curves. This integrating determines the area by representing the area under the graph of a variable as a trapezoid. The mean of the left and right sums is used in this rule. 

Whenever the fundamental value is continuous, the Trapezoidal Rule does not produce the same appropriate figure as Simpson's Rule. Simpson's Rule employs quadratic approximation rather than linear approximations. Both Simpson's Method and the Trapezoidal Rule provide an approximation estimate for the integrals, however Simpson's Rule provides a more precise approximation figure.

Check Important Notes for Heron’s Formula

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Trapezoid Rule Formula

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To calculate a numerical solution, we use the trapezoidal rule formula to calculate the region under the curve by dividing the overall area into small trapezoids instead of rectangles. The linear estimations of the functions are utilised to approximate the definite integrals in this rule. The mean of the left and right sums is used in the trapezoidal rule.

On [a, b], let y = f(x) be constant. The range [a, b] is divided into n equivalent subintervals, each with a width of h = (b - a)/n.

That is = x00 < x11 < x22 < ... < xnn = b

Here, 

y0y0, y1y1,y2y2…. are the estimates of function at x = 1, 2, 3….. respectively.

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Examples of Trapezoid Rule Formula

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Let's test it out by simulating the region under the function f(x)= 3 ln (x) with three trapezoids. When we label the first trapezoid T1, T2, and T3, here is how it looks in a diagram.

The area of trapezoid is \(h (\frac{b_{1}+b_{2}}{2})\) where h is the representation of height & ‘b’ as bases.

FINDING T1 AREA- 

Read More: Surface Area of a Right Circular Cone

By putting altogether, we get

By simplification,

FINDING T2-

Finding height & bases

By simplification,

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Riemann Sums

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Riemann summarises work on the concept of dividing the area under the arc into rectangular sections. The area gets closer and nearer to the present area as the count of rectangles grows. There is a functional f in the diagram below (x). This function divides the area into several rectangles. The total area underneath the curve is equivalent to the total of the rectangles' surfaces.

The right corner of the rectangles touches the curve in this case. Right-Riemann sums are the term for this. Left Riemann summation is formed when the left side of the rectangles contacts the curve in another scenario. The left-Riemann sum is illustrated in the diagram beneath.

The sum given will be

Check More: Factorial Formula

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Things to Remember

• The trapezoidal rule calculates the area of the zone under the graphs of the function f(x) that is approximated as a trapezoid.
• The trapezoidal rule can be thought of as the result of summing the left and right Riemann summation, and it is often defined in this way.
• By dividing the integral interval, implementing the trapezoidal method to each subinterval, and combining the results, the total can be estimated even better.
• In practice, "incorporating with the trapezoidal rule" usually refers to this "shackled" (or "composite") trapezoidal rule.
• The trapezoidal rule is part of the Newton–Cotes series of numerical integration equations, which includes the midpoint rule, which is related to the trapezoidal rule.
• Approaches with different basis points, including such Gaussian quadrature and Clenshaw–Curtis quadrature, are usually more effective for non-periodic distributions.
• Clenshaw quadrature can be seen as a shift in variables that allows arbitrary integrals to be expressed in terms of regular integrals, allowing the trapezoidal rule to be performed precisely.
• For the same amount of objective functions, the trapezoidal rule frequently provides sharper boundaries than Simpson's rule.

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