Numerical Integration (Trapezoidal Rule and Simpson's Rule)

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Step 1 - Data and Initialization:

You have a definite integral to compute: ∫[a, b] f(x)dx, where f(x) is the integrand, and a and b are the integration limits.

Step 2 - Choose a Method:

Select an integration method, such as the Trapezoidal Rule or Simpson's Rule, based on the desired level of accuracy.

Step 3 - Apply the Chosen Method:

Use the selected integration method to approximate the integral value. The formulas are as follows:

Trapezoidal Rule:

∫[a, b] f(x)dx ≈ h/2 [f(a) + 2f(a+h) + 2f(a+2h) + ... + 2f(b-h) + f(b)]

Simpson's Rule:

∫[a, b] f(x)dx ≈ h/3 [f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + ... + 2f(b-h) + 4f(b-2h) + f(b)]

Here, h is the step size, which determines the width of the subintervals used in the approximation.

Step 4 - Evaluate Accuracy and Convergence:

You can assess the accuracy of the numerical integration method by considering the error and comparing results obtained with different step sizes.

Step 5 - Output:

The result of the chosen integration method provides an approximation of the definite integral.

These methods are essential for approximating integrals when an analytical solution is not readily available.