5.5 Numerical Integration. concave down concave up concave down concave up concave down. - [PPT Powerpoint] (2024)

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Using integrals to find area works extremely well as long as we can find the antiderivative of the function. Sometimes, the function is too complicated to find the antiderivative. At other times, we don’t even have a function, but only measurements taken from real life. What we need is an efficient method to estimate area when we can not find the antiderivative.

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5.5 Numerical Integration concave down concave up concave downconcave up concave down Using integrals to find area worksextremely well as long as we can find the antiderivative of thefunction. Sometimes, the function is too complicated to find theantiderivative. At other times, we dont even have a function, butonly measurements taken from real life. What we need is anefficient method to estimate area when we can not find theantiderivative. Approximation of Area Under a Curve TrapezoidalRule: ( h = width of subinterval ) This gives us a betterapproximation than either left or right rectangles. The table belowrecords the outside temperature every hour from noon untilmidnight. What was the average temperature for the 12-hour period?Actual area under curve: Left-hand rectangular approximation:Approximate area: (too low) Approximate area: Right-handrectangular approximation: (too high) Averaging the two: 1.25%error (too high) Averaging right and left rectangles gives ustrapezoids: (still too high) Compare this with the Midpoint Rule:Approximate area: (too low)0.625% error The midpoint rule gives acloser approximation than the trapezoidal rule, but in the oppositedirection. Midpoint Rule: (too low)0.625% error Trapezoidal Rule:1.25% error (too high) Notice that the trapezoidal rule gives us ananswer that has twice as much error as the midpoint rule, but inthe opposite direction. If we use a weighted average: This is theexact answer! Oooh! Ahhh! Wow! This weighted approximation gives usa closer approximation than the midpoint or trapezoidal rules.Midpoint: Trapezoidal: twice midpointtrapezoidal Simpsons Rule: ( h= width of subinterval, n must be even ) Example: Simpsons rule canalso be interpreted as fitting parabolas to sections of the curve,which is why this example came out exactly. Simpsons rule willusually give a very good approximation with relatively fewsubintervals. It is especially useful when we have no equation andthe data points are determined experimentally. Example: UseSimpsons Rule with n=4 to approximate the value of. What is theerror? No error!! For most functions, Simpsons Rule gives a betterapproximation to an integral than the Trapezoidal Rule for a givenvalue of n. Sketch the graph of a function on a closed interval forwhich the Trapezoidal Rule obviously gives a better approximationthan Simpsons Rule for n = 4. Activity