5 3B Numerical Methods for estimating the area of an enclosed region. The Trapezoidal Rule for Approximating the Area Under a Closed Curve

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1 5 3B Numerical Methods for estimatig the area of a eclosed regio The Trapezoidal Rule for Approximatig the Area Uder a Closed Curve The trapezoidal rule requires a closed o a iterval from x = a to x = b. The iterval is the divided ito subitervals of width Δx = b a. A vertical segmet is draw from each of the x coordiates o the iterval to the poit where the segmet itersects the graph. The legth of each segmet will be f(x i ) for each of the x i o the iterval. Draw a straight lie segmet betwee each poit where the vertical segmets itersect the curve to create a series of trapezoids. f( x 4 ) f( x 0 ) f( x 1 ) f( x 3 ) f( x 2 ) f( x 5 ) f( x 6 ) x 0 x 1 x 2 x 3 x 4 x 5 x 6 To estimate the area uder the curve compute the area for each trapezoid ad fid their sum. You ca see that the sum of these areas are a close approximatio for the exact area uder the curve. The area of a trapezoid is A = h ( 2 base + base 1 2 ). The height of each trapezoid is Δx (the width of each iterval). The legth of ay base is the height of each vertical segmet. Legth of base i = f( x i ) The 2 bases of ay trapezoid are the legths of 2 cosecutive vertical segmets f( x i )ad f x i+1 ( ) f( x 0 ) is base of the first trapezoid. f( x 1 ) is base of the secod ad third trapezoid so it appears twice i the formula for the sums. Likewise the 3rd, 4th ad 5th bases also appear twice i the sums. The 6th base is oly the base of the last trapezoids so it appears oce i the formula. If we add the 6 trapezoids up we get the followig A = h ( 2 base + base 1 2 ) + h ( 2 base + base 12 3) + h ( 2 base + base 3 4 ) + h ( 2 base + base 4 5 ) + h ( 2 base + base 5 6 ) ( ) A = h 2 f(x )+ f(x ) 0 1 A = h 2 f(x )+ 2f(x )+ 2f(x )+ 2f(x )+ 2f(x )+ 2f(x )+ f(x ) ] Sectio 5 3B Lecture! Page 1 of 5! 2018 Eitel

2 The Trapezoidal Rule for Approximatig the Area Uder a closed curve The trapezoidal rule requires a closed curve o a iterval from x = a to x = b. The iterval is the divided ito subitervals of width Δx = b a The area of the x i 1, x i is A i = Δx ( 2 f ( x 1 i ) + f( x i )) The area of the trapezoids is A = h ( 2 base + base 1 2 ) + h ( 2 base + base 12 3) + h ( 2 base + base 3 4 ) h ( 2 base + base 1 ) 2 ( f(x 0 )+ f(x 1 )) + f(x 1 )+ f(x 2 ) ( ) + ( f(x 2 )+ f(x 3 )) ( base 1 + base ) 2 f(x )+ 2f(x )+ 2f(x )+2f(x )+...+ f(x )] ] trapezoid Example The example below uses f(x) = x 2 +1 from x = 0 to x = 2 ad 4 trapezoids Use the Trapezoid Rule to Approximate the area uder the curve f(x) = x 2 +1 from x = 0 to x = 2 with 4 itervals A =.5 2 f(0)+ 2f f(1)+2f + f(2) A = (2) A = ] = 19 4 = 4.75 Note: The picture makes it clear that the error i the trapezoidal rule estimate will deped o how cocave or covex is o each iterval: if is cocave up the trapezoidal rule will give a overestimate o that iterval, ad if is cocave dow the trapezoidal rule will give a uderestimate. Sice the secod derivative measures the cocavity of a fuctio, it is ituitively reasoable that the,maximum error should be proportioal to the secod derivative. Sectio 5 3B Lecture! Page 2 of 5! 2018 Eitel

3 Simpso s Rule This is the fial method we re goig to itroduce. Just like all the other techiques we divide the iterval a,b] ito subitervals. The width of each subiterval is, Δx = b a. However ulike the previous methods we eed to require that be eve. The reaso for this will be evidet i a bit. I the Trapezoid Rule we approximated the curve with a straight lie. For Simpso s Rule we are goig to approximate the fuctio with a quadratic fuctio o each iterval ad we re goig to require that the quadratic graph passes through three cosecutive poits The first iterval will use x o,x 1 ad x 2 ad the secod iterval will use x 2,x 3 ad x 4 Below is a sketch of this usig = 6. Each of the approximatios is colored differetly so we ca see how they actually work. Each approximatio actually covers two of the subitervals. This is the reaso for requirig to be eve. Some of the approximatios look more like a lie tha a quadratic, but they really are quadratics. Also ote that some of the approximatios do a better job tha others. It ca be show that the area uder the approximatio o the itervals x i 1, x i ] x i, x i+1 ] ad 3 f(x )+ 4f(x )+ 2f(x )+4 f(x )+2f(x ) f(x )+4 f(x )+1f(x ) ] The patter is 1, 4, 2, 4, 2, 4, 2,..., 4, 2,41, Notice that all the fuctio evaluatios at poits with odd subscripts are multiplied by 4 ad all the fuctio evaluatios at poits with eve subscripts (except for the first ad last) are multiplied by 2. If you ca remember this, this is a fairly easy rule to remember. The area i the figure above ca be approximated by 3 f(x )+ 4f(x )+ 2f(x )+4 f(x )+2f(x )+1f(x )+1f(x ) ] Sectio 5 3B Lecture! Page 3 of 5! 2018 Eitel

4 Example Approximate the area uder f(x) = 1 from x = 1 to x = 7 with 6 itervals. x +1 usig the Trapeziod Rule ad Simpso s Rule The graph is ot eeded but is provided below. Trapezoid Rule Δx = = 1 so the edpoits of the 6 itervals are x o = 1, x 1 = 2, x 3 = 3, x 3 = 4, x 4 = 5, x 5 = 6, x 6 = 7 ad for f(x) = 1 x +1 f( 1) = 1 2, f( 2) = 1 3,f( 3) = 1 4, f( 4) = 1 5, f( 5) = 1 6, f( 6) = 1 7, f( 7) = f(x )+ 2f(x )+ 2f(x )+2f(x )+...+ f(x ) ] Area! Area! Area! Exact Area = l(4)! Sectio 5 3B Lecture! Page 4 of 5! 2018 Eitel

5 Simpso s Rule Δx = = 1 so the edpoits of the 6 itervals are x o = 1, x 1 = 2, x 3 = 3, x 3 = 4, x 4 = 5, x 5 = 6, x 6 = 7 ad f( 1) = 1 2, f( 2) = 1 3,f( 3) = 1 4, f( 4) = 1 5, f( 5) = 1 6, f( 6) = 1 7, f( 7) = f(x 0)+ 4f(x 1 )+ 2f(x 2 )+4 f(x 3 )+2f(x 4 )+1f(x 5 )+1f(x 6 )] Area! Area! Area! Exact Area = l(4)! Sectio 5 3B Lecture! Page 5 of 5! 2018 Eitel