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MATLAB Trapezoidal Rule

Consider the function y=f(x) for the interval a≤x≤b, shown in figure:

To evaluate the definite integral, dx, we divide the interval a≤x≤b into subintervals each of length. Then, the number of points between x₀=a,and x_n=b is x₁=a+∆x,x₂=a+2∆x,…x_n-1=a+(n-1)∆x. Therefore, the integral from a to b is the sum of the integrals from a to x₁, from x₁ to x₂ and so on, and finally from x_n-1 to b.

The total area is:

0 P₁ x₁that is equal to plus the area of the trapezoid x₁ P₁ P₂ x₂ that is equal to , and so on. Then, the trapezoidal approximation becomes

Example

Using the trapezoidal rule with n=4, estimate the cost of the definite integral

Compare with the exact value and evaluate the percent error.

Solution:

The exact value of this integral is

For the trapezoidal rule approximation, we have

and by substitution into an equation

From equation 3 and equation 4, we find that the percent error is

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