10.
NumericalIntegration
LearningObjectives
- Use the TrapezoidalRule to solveproblems
- Estimateerrorsfor the TrapezoidalRule
- Use Simpson’s Rule to solveproblems
- EstimateErrorsfor Simpson’s Rule
Introduction
Recallthat we useddifferentwaysto approximatethe valueof integrals.TheseincludedRiemannSums
usingleft and right endpoints,as well as midpointsfor findingthe lengthof eachrectangulartile. In this lesson
we will learntwo othermethodsfor approximatingintegrals.The first of these,the TrapezoidalRule,uses
areasof trapezoidaltiles to approximatethe integral.The secondmethod,Simpson’s Rule,usesparabolas
to makethe approximation.
TrapezoidalRule
Let’s recallhow we woulduse the midpointrule with rectanglesto approximatethe area underthe
graphof from to
If insteadof usingthe midpointvaluewithineachsub-intervalto find the lengthof the correspondingrectangle,
we couldhaveinsteadformedtrapezoidsby joiningthe maximumand minimumvaluesof the functionwithin
eachsub-interval: