Calculus I · Unit 3A · lesson
Simpson’s Rule
Learn Simpson’s Rule through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Numerical and improper integrationWhat this section is building
Learn Simpson’s Rule through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Numerical rules replace a curve with simple local shapes; improper integrals replace a forbidden endpoint or infinite interval with a limit.
Choose the rule and partition, estimate scale and sign, or write the correct defining limit before evaluating.
Treating an approximation as exact, using Simpson's Rule with an invalid partition, or substituting infinity as though it were a number.
Learning objectives
Apply Simpson’s Rule with an even number of subintervals and understand why parabolic approximation is unusually accurate.
Simpson's Rule
Parabolas can approximate curvature better than lines
Simpson's Rule improves on straight-line approximation by fitting quadratic behavior across pairs of subintervals. This produces the characteristic weights and usually gives high accuracy for smooth functions. The method requires an even number of equal subintervals because each quadratic panel spans two of them.
The formula is easy to misuse if the data spacing or weight pattern is not checked first. Write the table of -values and weights before multiplying, and verify that the endpoints receive weight one. Simpson's Rule is still an approximation, not a symbolic antiderivative. Its strength is that it captures curvature efficiently, making it useful when function evaluations are available but exact integration is not.
For even ,
The alternating pattern is not optional bookkeeping; it comes from fitting quadratic pieces across pairs of subintervals.
Simpson’s Rule is exact for cubics
For on with , :
The exact integral is also .
u3a-simpson-01Use Simpson’s Rule with to estimate .
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Use values at 0, 1, and 2 with weights 1, 4, 1.
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Original instruction with traceable references.
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