Math 176 Calculus Sec. 5.1: Areas ad Distaces (Usig Fiite Sums) I. Area A. Cosider the problem of fidig the area uder the curve o the f y=-x 2 +5 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 5 equal subitervals of legth b a x =, i.e., each of width 2/5. [0, 0.4], [0.4, 0.8], [0.8, 1.2], [1.2, 1.6], [1.6, 2.0] The table below shows the values obtaied whe y (x) is evaluated at the correspodig poits. x y=-x 2 +5 0 5 0.4 4.84 0.8 4.36 1.2 3.56 1.6 2.44 2 1 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 2/5. 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. The area uder the curve, ca the be approximated by addig the areas of all the rectagles together.
Notice that whe usig the miimum values, i.e. 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. Let's calculate this estimate usig the right edpoits R 5 = 5 Â f x i Dx i 2 = f Ê ˆ 2 5 + f Ê 4ˆ 2 5 + f Ê 6ˆ 2 5 + f Ê 8 Ë ˆ 5 2 5 + f ( 2) 2 5 = 2 È 5 f Ê 2ˆ + f Ê 4ˆ + f Ê 6ˆ + f Ê 8 Ë ˆ Î Í 5 + f ( 2) = 2 4.84 + 4.36 + 3.56 + 2.44 +1 5 [ ] = 2 5 [ 16.2] = 6.48 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 oversetimate. Let's calculate this estimate usig the left edpoits 5 Â L 5 = f x i -1 i=1 Dx i = f ( 0) 2 5 + f Ê 2ˆ 2 5 + f Ê 4ˆ 2 5 + f Ê 6ˆ 2 5 + f Ê 8 Ë ˆ 5 2 5 = 2 È 5 f ( 0) + f Ê 2ˆ + f Ê 4ˆ + f Ê 6ˆ + f Ê 8 Ë ˆ Î Í 5 = 2 5 + 4.84 + 4.36 + 3.56 + 2.44 5 [ ] = 2 5 [ 20.2] = 8.08 From the two calculatios above we ca coclude that the area of the curve lies some where betwee the two approximatios, i.e. 6.48 < area of regio < 8.08
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. Let's calculate the above estimate: i.e. the Average or Midpoit Sum. 5 Â M 5 = f x i * Dx = 2 5 f Ê 1ˆ Á + 2 5 f Ê 3 Á ˆ + 2 5 f ( 1) + 2 5 f Ê 7ˆ Á + 2 5 f Ê 9ˆ Á = 2 È 5 f Ê 1 Á ˆ Ê 3ˆ Ê 7 + f Á + f ( 1) + f Á ˆ Ê 9ˆ Í + f Á Î = 2 4.96 + 4.64 + 4 +3.04 +1.76 5 [ ] = 2 5 [ 18.4] = 7.36 B. Thigs to ote: 1. The smaller the subitervals, the better the approximatio will be. This is because, the fuctio's values are chagig less i the subiterval, i.e. 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 5. Note that the error is miute compared with the previous work. a. Each of the above processes (lower sum, upper sum, midpoit sum) are just approximatios. They are ot exact. b. Whe you wat to calculate the Volume of a solid, you ca use similar techiques, oly you'll be usig rectagular solids or cyliders to approximate the volume.
2. Def : The area A of the regio S that lies uder the graph of the cotiuous f f is the limit of the sum of the areas of approximatig rectagles: A = lim R = lim f x i Æ Æ Â Dx = lim f x 1 Æ [ Dx + f ( x 2 )Dx +... f ( x )Dx], where Dx = b - a, x i = a + Dx i. a. Examples 1.) Use the Def above to fid a expressio for the exact area uder y=-x 2 +5 o the iterval [0,2]. 2.) Use the Def above to fid a expressio for the exact area uder f = x o the iterval [1,4]. C. Accuracy Error Magitude = true value - calculated value Relative Error = Percetage Error = true value - calculated sum true value 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 +5 over the domai [0, 2] is 22 = 7.33. Therefore the error associated with the approximatios 3 Error lowersum = 22 3-6.48 ª 0.11636364 22 3 are: Error upper sum = 22 3-8.08 22 3 ª 0.1018182 Error midpoit sum = 22 3-7.36 22 3 ª 0.0036364
II. Distace A. Costat Velocity If the velocity of a object remais costat, the the distace ca be computed by distace = velocity x time B. Variable velocity 1. If the velocity varies, i.e., the object moves with velocity, v=f(t) where a<t<b ad f(t)>0, the we will thik of the velocity as a costat o each subiterval. If the times are equally spaced, the Dt = b - a. Usig the left edpoits, the total distace= Â f ( t i -1 ) Dt. Usig the right edpoits, the total distace= Â f ( t i ) Dt. * Usig the midpoits, the total distace= Â f t i Dt. 2. This ca be thought of as fidig the area uder the velocity curve where the base of the rectagle is Dt = b - a ad the height of the rectagle is v=f(t). 3. Def : distace = A = lim f t i -1 Æ Â Dt = lim f t i Æ Â Dt i=1 C. Example 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 5 secods. t (s) v (m/s) t (s) v (m/s) 0 0 3.0 10.5 0.5 4.67 3.5 10.67 1.0 7.34 4.0 10.76 1.5 8.86 4.5 10.81 2.0 9.73 5.0 10.81 2.5 10.22 Usig left had edpoits
Usig right had edpoits Usig midpoits III. Additioal Example A. Ueve subitervals 1. 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 1 1 1.4 2.744 1.6 4.096 2.1 9.261 2.2 10.648 2.5 15.625 3 27 B. If the fuctio is ot strictly icreasig or decreasig.
1. 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 velocity (ft/s) 0 6.1 12.5 8.3 4.9 0