Sometimes it’s impossible for us to use integration formulas to find the exact value of a definite integral.
Consequently, we need to find approximate values for these integrals using numerical integration.
How?
By utilizing the Newton-Cotes Quadrature Formulas.
Similar to Riemann Sum approximation and Midpoint Rule, the Newton-Cotes formulas: Trapezoidal Rule and Simpson’s Rule are based on the strategy of changing a complicated function with an approximated function that is easy to integrate, and evaluating the integrand at equally spaced points in the interval.
From our previous study of Riemann Sums and Midpoint Rule we know that we subdivide the area under the curve into rectangles. Additionally, for the Trapezoidal rule we subdivide the region into trapezoids. But what about Simpson’s rule?
Well, Simpson’s rule, sometimes called Simpson’s 1/3 Rule, divides the region into parabolas (quadratics) for when the number of subintervals is even.
According to Wikipedia, Simpson’s Rule is a combination of the Midpoint and Trapezoidal approximation values, which is why we will see such incredible similarities between the formulas.
We will first look an example of how to use Simpson’s Rule to integrate a function given a Definite Integral.
Our second example we will use Simpson’s Rule and our knowledge of the Average Value function to estimate temperature given a table of values.
Lastly, we will compare all of our approximation techniques (i.e., Left and Right Riemann Sums, Midpoint Rule, Trapezoidal Rule, and Simpson’s Rule) and see how they are similar and different.
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