Trapezoidal rule

Part of engineering mathematics

1. Trapezoidal rule By Group 1

2. INTRODUCTION In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral. The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area.  

3. Basics for the Trapezoidal rule… The area of a trapezoid is given by the formula: where a,b are lengths of parallel base of trapezoid & h is the height Actual diagram of graph of function f(x) made up of trapezoids will look like the given figure  

4. Trapezoidal rule for Uniform Grid For a domain discretized Into N equally spaced panels b a

5. Trapezoidal rule for Non-uniform Grid When the grid spacing is not uniform, one can use the formula Where ( - )  

6. Error Analysis The accuracy with which the area under the actual curve is calculated will depend upon how closely the straight lines mimic the curve. The accuracy of representation can be increased by using a smaller interval h. Error= True value – Approximate value of integral % Error=  

7. 2 2 2 2 2 2 2 2

8. Continued…

9. Application For finding the area of irregular shapes. To find the Sectional areas In Earth work calculations trapezoidal rule is widely used to find the volume of Earth work In civil Engineering it has many applications especially in surveying To find the moment of intertia In naval architecture for approximation of the areas and volumes enclosed by the curves and surfaces.

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