For example suppose we divide the interval [0,2] into 5 equal subintervals of length

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1 Math 1206 Calculus Sec 1: Estimatig with Fiite Sums Abbreviatios: wrt with respect to! for all! there exists! therefore Def defiitio Th m Theorem sol solutio! perpedicular iff or! if ad oly if pt poit f fuctio eq equatio! is a elemet of st such that I Area A Cosider the problem of fidig the area uder the curve o the fuctio y!x 2 + over the domai [0,2] We ca approximate this area by usig a familiar shape, the rectagle If we divide the domai iterval ito several pieces, the draw rectagles havig the width of the pieces, ad the height of the curve, we ca get a rough idea of the total area For example suppose we divide the iterval [0,2] ito equal subitervals of legth!x b " a, ie, each width 2! 0 2 [0, 2], [ 2, 4 ], [ 4, 6], [ 6, 8], [ 8,2] The table below shows the values obtaied whe y(x) is evaluated at the correspodig pts x y!x 2 + x y!x Plottig these poits yields the followig graph If we fid the miimum value i the subiterval, ad use this as our height for that rectagle, we have what is kow as a iscribed rectagle See the graph below Now each of the above rectagles has the exact same width, amely 2 For this fuctio the height of each rectagle is give by calculatig the value of the fuctio at the right had edpoit of each subiterval Addig the areas of all the rectagles together ca approximate the area uder the curve

2 Notice that whe usig the miimum values, ie usig iscribed rectagles, we arrive at a estimate that is lower tha the actual area uder the curve Hece, this method results i what is kow as the lower sum or a uderestimate Lets calculate this estimate usig the right edpoits R! f x i " #x i f i1 2 ) " 2 + f 4 ( ) " 2 + f 6 ) " 2 + f 8 ) " 2 + f 2 2 * f 2, + 4 ( ( 2) / " 2 You ca also calculate a estimate usig the maximum value i the subiterval ad usig it as the height of the rectagles These rectagles are kow as circumscribed rectagles The resultig area approximatio will be greater tha the area uder the curve Cosequetly, we call this type of sum a upper sum or a overestimate Lets calculate this estimate usig the left edpoits L " f x i!1 # x i i1 f 0 # 2 + f , f 0 ( 2 ( * # 2 + f 4 ( ) * # 2 + f 6( * # 2 + f 8( * # 2 ( * + f 4 ( ) * + f 6( * + f 8( * 0 / From the two calculatios above we ca coclude that the area of the curve lies some where betwee the two approximatios, ie 648 < area of regio < 808

Aother method that ca yield a better approximatio is kow as the midpoit rule I the midpoit rule, you choose the value exactly i the middle of the subiterval to use i calculatig the height of the

3 Aother method that ca yield a better approximatio is kow as the midpoit rule I the midpoit rule, you choose the value exactly i the middle of the subiterval to use i calculatig the height of the rectagle; resultig i some rectagles beig both iscribed circumscribed Midpoits m 1 m 2 m 3 m 4 m Lets calculate the above estimate: ie the Average or Midpoit Sum M! f m i " #x i1 2 f 1 ) + 2 f 3 ) + 2 f 1 2 * f 1, ( + 2 f 7 7 ) + 2 f 9 ) 9 - ) / B Thigs to ote: 1 The smaller the subitervals, the better the approximatio will be This is because, the fuctios values are chagig less i the subiterval, ie, the value of the fuctio is fairly costat i each subiterval Cosequetly, we are ot approximatig by such a rough amout each time For example, here is the same regio divided ito 20 rectagle istead of Note that the error is miute compared with the previous work 2 Each of the above processes (lower sum, upper sum, midpoit sum) are just approximatios They are ot exact 3 Whe you wat to calculate the Volume of a solid, you ca use similar techiques, oly youll be usig rectagular solids or cyliders to approximate the volume

4 C Accuracy Error Magitude true value - calculated value Relative Error true value! calculated sum true value Percetage Error true value! calculated sum true value ( 100) For the example i part I, the true value for the area uder the curve y!x 2 + over the domai [0, 2] is Therefore, the error associated with the approximatios are: Relative Error lower sum 22 3! " Relative Error upper sum 22 3! " Relative Error midpoit sum 22 3! " II Distace A Costat Velocity If the velocity of a object remais costat, the the distace (velocity)(time) B Variable velocity 1 If the velocity varies, ie, the object moves with velocity, vf(t) where a<t<b ad f(t)>0, the we will thik of the velocity as a costat o each subiterval If the chages i time are equally spaced, the!t b " a Usig the left edpoits, the total distace f ( t i!1 ) " #t Usig the right edpoits, the total distace i 1 # f ( t i )! "t i 1 Usig the midpoits, the total distace! f m i " #t i1 2 This ca be thought of as fidig the area uder the velocity curve where the base of the rectagle is!t b " a ad the height of the rectagle is vf(t)

5 C Examples 1 A radar gu was used to record the speed of a ruer at the times i the table Estimate the distace the ruer covered durig those miutes t(mi) v ( m hr ) t(mi) v ( m hr ) t(mi) v ( m hr ) Usig left had edpoits Usig right had edpoits Usig midpoits 2 Ueve subitervals Give y x 3 o [1,3] Use the table below to estimate the area betwee the curve ad the x-axis usig the left edpoits x y x

3 If the fuctio is ot strictly icreasig or decreasig The table below gives the velocity at the specified time Use this data to give a uderestimatio of the distace traveled time (s) 0 2 4 6 8 10

6 3 If the fuctio is ot strictly icreasig or decreasig The table below gives the velocity at the specified time Use this data to give a uderestimatio of the distace traveled time (s) velocity (ft/s) III Displacemet vs Distace Traveled Give a body o a coordiate lie with positio f, s(t) ad velocity f, v(t) o the time iterval, a! t! b the A Displacemet 1 Def: Displacemet is the differece betwee its iitial ad fial positios,! s s b " s( a) 2 Displacemet is the chage i positio, NOT total distace B Total Distace Traveled 1 Without directio chage: Total distace traveled! s s( b) " s( a) 2 Directio chages: If the directio of the body chages oe or more times, the total distace traveled is approximately the sum v(t 1 ) (!t) + v(t 2 ) (!t) + v(t 3 ) (!t) + + v(t ) (!t) " v(t i ) (!t), where v(t) the body s speed i1 IV Average Value of a Noegative Fuctio A Fiite collectio of values f ave sum of the umbers B Cotiuous Fuctio o the iterval [a,b] 1 Costat fuctio, f (x) c f ave c c y c a b

7 2 Nocostat Fuctio, y g(x) g ave 1 (Area beeath its graph) iterval legth g ave 1 (Midpoit Sum) (b! a) C EXAMPLES 1 Fid the average value of f f x subitervals (see earlier example to area)!x 2 + o [0,2] usig 2 Fid the average value of f f ( x) x o [2,4] usig 4 subitervals

For example suppose we divide the interval [0,2] into 5 equal subintervals of length

For example suppose we divide the interval [0,2] into 5 equal subintervals of length Math 120c Calculus Sec 1: Estimatig with Fiite Sums I Area A Cosider the problem of fidig the area uder the curve o the fuctio y!x 2 + over the domai [0,2] We ca approximate this area by usig a familiar