Simpson's Rule, Single Variable Calculus

Let's look at Simpson's Rule to estimate the value of the definite integral of y=f(x) from x=a to x=b (with lots of pictures). We also discuss how to bound the size of the estimation error. Computational supplement (in MATLAB) here:    • Simpson's Rule MATLAB Supplement, Single V...   This lesson introduces Simpson's rule, a numerical method to estimate the value of a definite integral. It's particularly useful for functions that are difficult or impossible to anti-differentiate by hand. The method approximates the area under a curve using parabolic segments over an even number of sub-intervals. We begin with visual examples, showing how Simpson’s rule works by fitting parabolas through groups of three points and estimating the area under each parabola. We see how increasing the number of sub-intervals improves accuracy. Next, the formula for Simpson’s rule is derived by integrating a general parabola and relating the area to function values at the interval points. Finally, the computation is applied to specific examples, including a discussion of how to assess the error of the approximation. The lesson also covers a theorem to bound the error of Simpson's rule based on the fourth derivative of the function, showing that the error decreases as the number of sub-intervals increases. An exercise demonstrates how to determine the number of sub-intervals needed to guarantee an approximation within a certain accuracy. #mathematics #calculus #integration #simpsonsrule #numericalmethod #numericalmethods #DefiniteIntegral #IntegralApproximation #AppliedMathematics #calculus2 This video is part of my full Single Variable Calculus II course playlist (Calc 2, MA 241 at NC State University):    • Single Variable Calculus II - Complete Sem...   #calculus2 #singlevariablecalculus