Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl Theorem of Clculus, interpreted in some pplied context), plus the ones we review in clss, including verge vlue, volumes by slicing, work s the integrl of force, nd possibly others lter in the term. However, it s worth thinking bout ll of them, prticulrly for those of you who re tking clculus in order to use it in engineering or some other subject. The Net Chnge Theorem (for functions of time): This is just the Fundmentl Theorem of Clculus, pplied. The Fundmentl Theorem of Clculus tells us tht, if F hs continuous derivtive on the intervl [, b], then F (t) dt = F (b) F (). If t represents time, nd F (t) represents some quntity tht chnges over time, then F (t) represents the rte of chnge of F (t) (with respect to time), nd F (b) F () represents the net chnge in F (t) between times t = nd t = b. ( Net chnge mens tht we llow decreses nd increses to cncel ech other out. For exmple, if the temperture is 5 degrees t 7 AM, rises to 3 degrees t 2 PM, nd then drops to degrees t 7 PM, then between 7 AM nd 7 PM the temperture rises by 28 degrees nd then flls by 2 degrees, nd the net chnge is 26 degrees; positive net chnge denotes n increse. ) This is wht the Net Chnge Theorem sys. Quoting directly from our textbook: Net Chnge Theorem. The integrl of rte of chnge is the net chnge: F (x) dx = F (b) F (). I got these numbers from the Wether Underground forecst for Jnury 8, 205 in Hnover, New Hmpshire; degrees re in Fhrenheit.
Pges 324 nd 325 of the textbook list severl instnces of the Net Chnge Theorem. In prticulr, the Net Chnge Theorem tells us tht the integrl of velocity is distnce: For n object trveling long stright pth (with positive nd negtive directions chosen), if F (t) denotes the distnce from the strting point t time t, then F (t) denotes velocity, the rte of chnge of distnce with respect to time, nd the integrl of the velocity between times t = nd t = b F (t) dt denotes the net distnce trveled between times t = nd t = b. Averge Vlue: The verge vlue of function f(x) for x in the intervl [, b] is given by b f(x) dx. This is very like the formul for computing the verge of finitely mny numbers: To find the verge of finite set of numbers, dd up the numbers, nd divide by the size of the set. To find the verge vlue of function f(x) on n intervl, use n integrl to totl up the vlues of f, nd divide by the length of the intervl. We could pproch this more formlly: To pproximte the verge vlue of f(x) on the intervl [, b], divide the intervl [, b] into lrge number of equl subintervls (sy n subintervls), choose point x i in the i th subintervl for i =, 2,... n, nd tke the verge of the vlues f(x i ). This gives us n f(x i ) = b b n b f(x i ) = b f(x i ) x, f(x i ) b n = where x = b is the size of the subintervls. This sum is just the n Riemnn sum tht we used to pproximte the re under curve. When we compute the verge s the limit of closer nd closer pproximtions s 2
n, we get b lim n f(x i ) x = b f(x) dx. Note: Forml computtions, such s the derivtion of the formul for verge vlue bove, re not of interest only to theoreticl mthemticins. If you re using clculus in engineering or economics or medicl reserch, you don t wnt to be limited to the pplictions of integrtion on some list; you wnt to know how to recognize new ppliction of integrtion in the wild, so to spek. You recognize potentil ppliction of integrtion by seeing tht you cn pproximte the thing you re interested in s sum of smll pieces, nd get better pproximtion by using lrger number of smller pieces. Then, some computtion like this is needed to tell you exctly wht integrl you should use to compute tht thing you re interested in. In this course, we won t expect you to derive these formuls, such s this formul for verge vlue, just to pply them. However, in physics or engineering course, you my well hve to do problems such s some of the computtions of work in the section on other pplictions, tht essentilly require you to figure out the pproprite integrl in this wy. The following sections will sometimes give these forml computtions, nd sometimes give more intuitive rguments. You might try to provide the forml computtion when it is not included. Volumes by Slicing: To find the volume of solid object occupying the region in spce between x = nd x = b, where x is mesured long some stright xis, integrte the object s cross-sectionl re between x = nd x = b. More precisely, suppose A(x) is the cross-sectionl re t x; tht is, the cross-sectionl re we get if we slice the object t the point x, perpendiculrly to the xis. Then the volume of the object is given by V = A(x) dx. We cn see why this is the cse just s we did with the formul for verge vlue: We cn pproximte the volume of the object by cutting it, perpendiculr to the x-xis, into mny thin slices of thickness x. The volume of ech thin slice cn be pproximted by the cross-sectionl re 3
A(x i ) t some prticulr point, times the thickness x of the slice. Then we cn pproximte the volume of the object by dding up these pproximte volumes A(x i ) x of the slices, nd find the exct volume by tking the limit s the number of slices pproches infinity. We cn lso give n informl rgument tht this mkes sense. If the cross-sectionl re is the sme t ny point, the volume of the object is the product of its length nd its cross-sectionl re. For exmple, cylinder of length h nd cross-section circle of rdius r hs volume V = πr 2 h, formul tht my be fmilir. It mkes sense tht if the cross-sectionl re vries, you could possibly get the volume by multiplying the length by the verge cross-sectionl re. Using the formul for verge vlue, we get V = (b ) b A(x) dx = A(x) d. Work: In physics, if force of mgnitude F cts on n object s it moves distnce d (where the force is cting in the direction of the object s motion), the work done by tht force on tht object is the product F d. Informlly, we sy work equls force times distnce. If the force is negtive, tht indictes tht the force is cting opposite to the direction of motion, nd the work F d is lso negtive. Intuitively, positive work is work done by force cting in the direction of motion, nd negtive work indictes tht work must be done to oppose force cting opposite to the direction of motion. If the force cting on the object is different t different points long its pth, then we compute work using n integrl: Suppose n object moves long the x-xis from point x = to point x = b, nd force on the object cting in the x-direction hs mgnitude F (x) t point x. (Agin, if F (x) is negtive, tht indictes tht the force is cting in the negtive x-direction.) Then the net work done by tht force on the object is given by the integrl of F (x) over the intervl of motion: W = F (x) dx. To see why this is the cse, consider pproximting the work: Divide the intervl of motion [, b] into lrge number n of equl subintervls of length 4
x = b. If n is lrge enough, x will be smll enough tht the force n doesn t chnge very much over ny one subintervl, nd we cn pproximte the force in the i th subintervl by the force F (x i ) t ny one point x i in tht subintervl. Then, becuse work equls force times distnce, the work done s the object moves long the i th subintervl is pproximtely F (x i ) x. We pproximte the work over the entire intervl by dding up the pproximte vlues of the work over the subintervls: W F (x i ) x. We cn get closer pproximtion by tking lrger number of subintervls, nd n exct vlue by tking the limit s n pproches infinity: W = lim n F (x i ) x. We cn recognize this limit s the definition of the definite integrl: W = F (x) dx. Just s with volumes by slicing, we could lso give n informl rgument tht this mkes sense, becuse this is the distnce times the verge force. However, our intuition cn misled us in rel life, ll too mny things tht mke sense turn out not to be true nd therefore the resoning with pproximtions nd limits is not only more forml but lso more relible. Units of Work: We write other quntities in terms of the bsic quntities of mss, distnce, nd time. Newton s Second Lw (force equls mss times ccelertion) tells us the units of force should be the units of mss times the units of ccelertion. In the SI system, force is mesured in Newtons (N), defined by N = kg m/s 2, where kg is kilogrm, m is meter, nd s is second, nd work is mesured in Joules (J), defined by J = kg m 2 /s 2. 5
Other Applictions: Another stndrd ppliction is finding totl mss by integrting mss density, totl chrge by integrting chrge density, nd so forth. For exmple, suppose wire occupying the portion of the x-xis x b hs vrible mss density of ρ(x) grms per meter t point x. If the mss density were constnt, we would just multiply the mss density (in grms per meter) by the totl length (in meters) to get the totl mss (in grms). Using n intuitive rgument, we cn compute the totl mss s the length times the verge mss density, or (b ) b ρ(x) dx = ρ(x) dx. Computing work vi n integrl is often not s strightforwrd s integrting force over some intervl of distnce. For exmple, suppose chin of length l nd constnt mss density ρ is hnging by one end from the top of wll, nd we wnt to compute the work required to pull the entire chin up to the top of the wll. The force of grvity on n object of mss m is mg, nd this is the force we must do work ginst. More work is required to pull the lst prt of the chin up thn the first prt of the chin, since the lst prt of the chin must cover longer distnce. Letting x be the distnce from the top of the wll, the chin occupies the intervl 0 x l. We divide the intervl into n subintervls of length x = b n. Choose x i in the i th subintervl. The portion of the chin occupying the i th subintervl hs mss ρ x, nd must be moved distnce of pproximtely x i, so the work done on this portion of the chin is pproximtely the force on tht portion of the chin (ρ( x)g) times the pproximte distnce it must be moved, x i, or ρgx i x. We pproximte the work on the entire chin by dding up the work done on ech smll subintervl, nd find the ctul vlue by tking limit s n pproches infinity: lim n ρgx i x = l 0 ρgx dx. You cn find other exmples of computing work, nd other pplictions of integrtion, in the textbook. 6