Laboratory I.10 It All Adds Up

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1 Laboratory I. It All Adds Up Goals The studet wll work wth Rema sums ad evaluate them usg Derve. The studet wll see applcatos of tegrals as accumulatos of chages. The studet wll revew curve fttg sklls. Before the Lab The Before the Lab ad Ready for Lab sectos are loger tha usual for ths lab; be sure to start early. (O the other had, the I the Lab ad After the Lab are correspodgly shorter.) Part I: The Ital Scearo The Evrometal Protecto Agecy recetly vestgated a spll of radoactve ode. Measuremets showed the ambet radato levels at the ste to be four tmes the maxmum acceptable lmt of.6 mllrems/hour (abbrevated mrems/hr). The EPA ordered a evacuato of the area. You wll vestgate several aspects of ths cdet usg tegrals ad other basc formato about fuctos. At the begg of the vestgato, the emsso rate was measured to be 2.4 mrems/hr. Over the frst four hours, the followg data were recorded: There are some questos o ths scearo the Ready for Lab? secto. If you eed ay help to aswer those questos, please study the questo ad aswers o the last page. Part II: Rema Sums ad Ther Applcato ths Lab. I addto to kowg the emsso rate of the radato, the EPA s also cocered wth the total radato. You ca use the table above to estmate the total radato ay oe hour perod as follows. From the table of values we ca assume that the radato rate s gettg smaller wth tme. For the frst hour, you kow the radato rate was at most 2.4 mrems/hr ad at least mrems/hr. So you kow that the total radato emtted that frst hour was betwee 2.4 ad mrems. Note that the upper-estmate of 2.4 mrems was based o the readg at the start of the frst hour, or the left edpot of the tme terval [,]. Correspodgly, the lower-estmate of mrems was based o the readg at the ed of the frst hour, or the rght edpot of the tme terval [,]. The accumulato of the radato emtted all the dvdual hours s the total of all the radato over a loger tme terval. We wll estmate the total radato over the terval from tme to tme 4, e, over [,4]

2 For each oe-hour terval, there s a left estmate of the radato emtted that hour based o the radato emsso rate at the left-had edpot of that terval ad a rght estmate of the radato emtted based o the rght-had edpot of that tme terval. The total of the left estmates over the total terval [,4] s called the left Rema sum (or left-had sum, or LRS). The total of the rght estmates over the terval [,4] s called the rght Rema sum (or rght-had sum, or RRS). I the table below, record the left- ad rght-estmates for each oe hour terval ad the LRS ad RRS for the total 4 hour terval. Do t tur ths table, but be sure you are cofdet you have flled t out correctly. Some values are gve to help you cofrm ths. Tme Iterval Left Estmate Rght Estmate/ RRS Legth of the tme terval [,] [,2] [2,3] [3,4] [,4] f ( t 2.4 f ( t f ( t f ( t Left Rema Sum 3 LRS: f ( t Δt = ) Rght Rema Sum RRS: Δ t = 4 Yes, there s a dfferece betwee the LRS ad RRS for the terval [,4]. The reaso for ths s that we assume the emsso rate to be costat over each hour terval. It was t. To refe our aswer, we could have take smaller tervals ad emsso rate readgs more ofte. Dvdg the terval [,4] to smaller subtervals.e.makg Δ t smaller would requre more subtervals. Takg the lmt of the RRS ad LRS as Δ t goes to zero gves the exact value of the total emtted radato. Ths accumulato fucto s called the defte tegral of the radato rate Part III. Rema Sums Derve You may ot yet have bee troduced to Rema sums, whose lmt s the defte tegral. Ths s dsplayed o page 38 of the textbook. The defte tegral of f(t) over the terval [a,b] s defed as the lmt of the (left or rght) Rema sums as the legth of the subtervals teds to zero. b f ( t) dx = lm a f ( t * ) Δt where Δt s the legth of each tme terval we use, the example above Δt = ad f ( t * ) s the estmate the -th terval. We ca wrte the left sum ad the rght sum as o the ext page: MATH 243 Lab Page 2 of 9

3 Left Rema sum Rght Rema sum where Δ t = (legth of the whole tme terval)/ Now, both of the sums above have the same lmt as goes to fty. As goes to fty, zero. The bgger gets, the closer the Rema sum s to the exact value of the tegral. The purpose of ths Before the Lab secto s to work through the traslato of the algebrac summato otato ad To calculate a Rema sum we (ad Derve) eed to kow four thgs: what s the fucto f? what are the edpots a ad b? ad what s, the umber of terms the sum? Δt goes to MATH 243 Lab Page 3 of 9 to Derve s verso of the same thg. We frst eed to re-wrte t ad Δt terms of a, b ad. Δt s the dstace from a to b, cut to peces, so that s Δt = (b a)/. To uderstad t, let s look at the patter we see: t = a t = a + Δt = a + Δt = a + (b a)/ t 2 = t + Δt = a + 2 Δt = a + 2(b a)/ t 3 = t 2 + Δt = a + 3 Δt = a + 3(b a)/ etc. See the patter? t = a + (b a)/. Substtutg our ew formulas for x ad Δx to the rght had sum formula, we get: = b a b a f a + Our goal was to wrte the Rema sum just terms of f, a, b, ad, ad we have. Whe you get to the lab, you ca type the above Rema sum two les: Author f(x) := the Author RRS(a, b, ) := sum(f(a + (b a)/)(b a)/,,, ) The frst le, f(x) :=, tells Derve that the letter f wll stad for a fucto of x the ext formula. The secod le creates a fucto called RRS ( = Rght Rema Sum) whch, gve the ecessary gredets of f, a, b, ad, creates the exact sum we see above. Suppose we wated to use ths lab to estmate 5 2 x 3 dx wth = 7 subtervals. The, after dog the above, we would:

4 Author f(x) := x^3 Author RRS(2, 5, 7) Approxmate the result. If you try ths lab, you should get So, there s a Ready for Lab? questo for ths secto as well. Questo 6 Ready for Lab s about the LRS formula. Eter the formula you got for that questo to Derve ad the Author LRS(2,5,7) ad approxmate the result ad you should get Do ot cotue workg o the lab f the aswer s ot correct. Get help from your TA to troubleshoot the formula. I the Lab (8 pts). Author R(t):= your formula for the ode data, as well as the RRS ad LRS formulas. Be sure to use R, stead of f, your RRS ad LRS formulas, sce f you use f, Derve wll look for a fucto called f ad ot fd t. (You wll fd a formula for R(t) the Before the Lab secto of the lab.) 4. a) Use RRS ad LRS wth = 4 to estmate the tegral R ( t) dt from t = to t = 4. Fll the approprate les Table. Sce you kow that the exact tegral s somewhere betwee the two. Estmate the maxmum error you could make f you used the average of the two estmates to evaluate the tegral, b) We wll see class that creasg creases the accuracy of the estmate you get from Rema sums. Cotue fllg out les o the table - you wo t have to use them all - utl your estmated maxmum error s less tha.25. c) Use Derve s Calculus Itegrate commad, followed by Approxmate, to evaluate 4 R( t) dt exactly. (I the Itegrate dalog box, you ca use R(t) for the fucto; you re dog a defte tegral, ad the upper ad lower bouds are 4 ad respectvely.) Fll the blak below Table. Were you rght to stop where you dd the table? That s, s the average of your last row wth.25 of the exact value? 2. a) I Ready for Lab? #4, you foud the tme whe the ode radato levels retur to the acceptable level of.6 mrem/hr. Call that tme ta. Use Calculus Itegrate to evaluate t a R(t) dt (Of course, whe you eter ths to Derve, you ll use your actual value for ta, ot the symbols MATH 243 Lab Page 4 of 9

5 ta.) Fll the approprate spot Table 2. You wll be asked about the meag of ths tegral a After the Lab questo. b) Cotue fllg rows of Table 2 - aga, you wo t use them all - utl the value you get s less tha. The add all the tegrals you foud ad fll the blak below Table 2. I the After the Lab secto aga, you wll be asked about the meag of the sum. c) Fally, you ca, for some tegrals, you ca use fty ( ) as a boud. Use Calculus Itegrate to evaluate R(t) dt (Clck o the key the Itegrate dalog box whe you are eterg the upper boud.) Table Table 2 Sum of tegrals above Ready for Lab? ( pts) R(t) dt= MATH 243 Lab Page 5 of 9

6 Questos o the tal scearo:. Based o the umbers the table, calculate a upper ad lower boud for the total amout of radato ( mrems) emtted the frst four hours. (LRS ad RRS) 2. Let R(t) deote the radato level mrems/hr at tme t hours after the ode spll. Wrte a tegral that represets the total amout of radato ( mrems) emtted the frst four hours. (Warg: t s easer tha t looks.) 3. If the EPA reled solely o measuremets, t would have to keep measurg every hour to see whe t was safe to retur to the ste of the spll. It would be better f there was a formula whch predcted radato levels at future tmes. There s!. Radato levels decay expoetally, so R(t) ca be wrtte as R e kt. Fd R ad k for the data the table. 4. Based o your formula from #3, whe wll the radato level fall to the EPA s estmate of a acceptable radato level of.6 mrems/hr? 5. Based o the data the Table for radato, s the left Rema sum gog to be a uderestmate, or a overestmate? Why? Is the rght Rema sum gog to be a uderestmate, or a overestmate? Why? 6. Wrte dow a Derve formula for the Left Rema Sum, based o what was the Rema Sum secto of Before the Lab. After the Lab ( pts). From Table, by what factor does doublg the umber of rectagles seem to reduce the possble error? 2. What s the meag of the umber you calculate each of parts (a), (b), ad (c) of I the Lab #2? MATH 243 Lab Page 6 of 9