Math 1314 Lesson 16 Area and Riemann Sums and Lesson 17 Riemann Sums Using GeoGebra; Definite Integrals
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1 Math 1314 Lesso 16 Area ad Riema Sums ad Lesso 17 Riema Sums Usig GeoGebra; Defiite Itegrals The secod questio studied i calculus is the area questio. If a regio coforms to a kow formula from geometry, the fidig the area is ot difficult; simply determie the dimesios ad apply the appropriate formula. Suppose we wat to fid the area of a regio that is ot so icely shaped. For example, cosider the fuctio show below. The area below the curve ad above the x axis caot be determied by a kow formula, so we ll eed a method for approximatig the area. Suppose we wat to fid the area uder the parabola ad above the x axis, betwee the lies x = ad x = -. We ca approximate the area uder the curve by subdividig the iterval [-, ] ito smaller itervals ad the draw rectagles extedig from the x axis up to the curve. Suppose we divide the regio ito two parts ad draw two rectagles. We ca fid the area of each rectagle ad add them together. That will give us a approximatio of the area uder the curve. This method is called fidig a Riema sum. This would ot give a very good approximatio, as a large regio i Quadrat will be left out i the approximatio of the area, ad a large regio i Quadrat 1 will be icluded ad should ot be. Now suppose we icrease the umber of rectagles that we draw to four. We ll fid the area of each of the four rectagles ad add them up. Here s the graph for this situatio. 1
2 The approximatio will be more accurate, but it still is t perfect. Let s icrease the umber of rectagles to 8: As we add more ad more rectagles, the accuracy improves. We re still ot to a exact area, but the area we d fid usig more rectagles is clearly more accurate tha the area we d fid if we just used rectagles. Suppose we let the umber of rectagles icrease without boud. If we do this, the width of each rectagle becomes smaller ad smaller, as the umber of rectagles approaches ifiity, there will be o area that is icluded that should t be ad oe left out that should be icluded. This process is beyod the scope of this course, so we will limit the umber of rectagles i the problems we work to a fiite umber. Usig left edpoits is ot the oly optio we have i workig these problems. We ca also use right edpoits or midpoits. The first graph below shows the regio with eight rectagles, usig right edpoits. The secod graph below shows the regio with eight rectagles, usig midpoits. Right Edpoits Midpoits Now, how do we approximate the area?
3 1. Start by fidig the width of each rectagle. We ca compute the width of the rectagles usig this formula: b a x I this formula, a ad b are the edpoits of the iterval [a, b] ad is the umber of rectagles. x stads for the chage i x.. Now fid the height of the rectagles. Subdivide the iterval ito subitervals, each of width x. Use the appropriate poit i each subiterval to compute the value of the fuctio at each of these poits (gives the heights of the rectagles).. 3. Fid the area of each rectagle ad add them up. A f x1 f x... f x x To get a exact area, we would eed to let the umber of rectagles icrease without boud: A lim f x1 f x f x x This last computatio is quite difficult, we will ot work problem of this type. Istead, we will use a limited umber of rectagles i the problems that we work. The process we are usig to approximate the area uder the curve is called fidig a Riema sum. These sums are amed after the Germa mathematicia who developed them. Example 1: Let f ( x) x 1. Approximate the area uder the curve, usig 4 subdivisios, o the iterval [0, ] usig left edpoits. Now suppose the fuctio, iterval ad/or subdivisios we wish to work with are ot so ice. You would ot wat to work this type of problem by had. We ca work Riema sum problems usig GeoGebra. The commad is: RectagleSum[<Fuctio>,<Start x-value>,<ed x-value>,<number of rectagles>,<positio for rectagle start>] 3
4 Positio of rectagle start : 0 correspods to left edpoits, 0.5 correspods to midpoits ad 1 correspods to right edpoits. Example : Approximate the area betwee the x axis ad the graph of 3 f ( x) 0.3x 0.807x 3.5x o [-.8, 1.33] with 50 rectagles ad midpoits. Eter the fuctio i GGB. Aswer: Example 3: Approximate the area betwee the x axis ad the fuctio 15 f ( x) o [0.075, 8.1] usig 1 rectagles ad left edpoits x 19.17x 1 Eter the fuctio i GGB. Aswer: Upper ad Lower Sums Usig GeoGebra You ca also fid a related quatity usig GeoGebra, the upper sum ad/or the lower sum. Rather tha always usig the left edpoit, the right edpoit or the midpoit of the iterval to fid the height of the rectagle, the upper sum uses the biggest y value o each iterval as the height of the rectagle ad the lower sum uses the smallest y value o each iterval as the height of the rectagle, o matter where o the iterval that value occurs. The commad for Upper Sum is: UpperSum[<Fuctio>,<Start x-value>,<ed x-value>,<number of Rectagles>] 4
5 The commad for Lower Sum is: LowerSum[<Fuctio>,<Start x-value>,<ed x-value>,<number of Rectagles>] Example 4: Use GeoGebra to fid the upper sum ad the lower sum for iterval [-3, 5] usig 35 rectagles. Eter the fuctio i GGB. 1 x f x e o the Upper Sum Aswer: Lower Sum Aswer: The Defiite Itegral Let f be defied o [a, b]. If [ f ( x ) f ( x )... f ( x )] x lim exists for all choices of 1 b a represetative poits i the subitervals of [a, b] of equal width x, the this limit is called the defiite itegral of f from a to b. The defiite itegral is oted by b a ( = lim ( 1) ( )... ( ) f x) dx f x f x f x x. The umber a is called the lower limit of itegratio ad the umber b is called the upper limit of itegratio. 5
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