DEFINITIONS INTERPRETATIONS • !leuristies The defi nite integral captures the idea of addin g the va lues or a run clion over a continuulll • Riemann sum A su itably we ig hted sum of values A de fin ite illlcgra l is the li mi t ing value of sueh sums A Ri e man n s um of a funct ion f de f incd on [II ,h] is de te rmincd by a partitio n , which is a fini te division of [{I h·1 into subintervals, ty p ically exprcssed by {I =xu I) Ml~ ~~ ~~~~~~B""~I~ ~ -b a unspecifi ed ) X- dx =! x ' - + C j '!x' - i Th e consta nt C, w hich may have any real valuc, is the constant of integr a tion (Com puter prog ms, and this cha rt may om it the constant, it being understood by the kn owledgea hle user that the g iven a nl idcrivative is just one representative of a m ily.) is f ; and if (x) ldx • Fundamental theorem of calculus O ne part o f the theore m is use d to evaluate integrals: Iffis continuo us o n [(I, h] a nd A is a n ant idcr iva ti ve off n n th at inte rval , then j ·"f (x) dx=A(x )I"C= A (b) - A(a) (, {/ The other part is used to construct antide rivativcs: If f is conti nu ous o n [a , h) th en the fun cti on A (x)l= ( j U)d l is an a ntiderivative of f on [(I ,h] : a ~ b apprehend or cvaluate In effect, the "a rea" is smoothly rcd istributed w it hout changing the integral;' value !I' g is a fun ction w ith cont inuous deriva tive andf i, continuous then " lI )dn = I'df (.9(f ))g ' (f )dl, 1"f( where c d are points with g (C)= il a nd g(rI )= h In pract ice, su bstit !!!r lI = g(l); compu te t11/ = g'(I)dl; and find what t is when 1/ = and I/ = h E.g I/ =sin t e ffe cts the transrorm ation I> ( ) Iff is non negative , then (".f x dx is no nnegati ve i[ ' j(x )dx i< t II!I / ,' lnnnunuu : ~ ~ whkh becomes X - {I , an exp ression for the fam ily or n integra tion can be cha nged to make an in tegral eas ier to Use this in tegra l evalu at ions w ith rough ove restimates o r u nderes tima tes l/.hJ, the n so , • Cbange of variable formula An integrand and limit, of " to ch ec k U C is attained i\1VT for Integra h L · (b -alS l "f (xldX S, M (/)- a) Iff is int egrablc on f somewhere on the inLe rval : -b - -a a f(x) dx =.f( ; ) \ -u- du= I' " "~ ! - s in-I co~ I til /.", ,; of of on · I Oil by 2a- I , T he indefinite in tegra l ora runction! ' dx In thc case g - I , the a\eragc value of • Integrability & inequalities A co ntinuo us fun cti on o n a closed inter val is integra bl e Integ rab ility On [tI h] imp li es int egra bi lity on c losed sub inte rva ls of [tI h] Ass umin gfi s in teg rable , if L "'f(x) M for a ll x in [{I ,hJ, th e n ~ IIiII ·Mean value theorem for integrals Ir f an d ~ a rc conti nuo us o n [II ,h], then there is a S in [a.h] such Lhat f«, L_ in\o l vin g a THEORY II c OI11[l osition A( u )= ( "f (tldt T o d i frcre nt ia te, u sc the c hai n rule x~ • Definite integral The definite inte gra l off frol11 {I to b may bc described as is a and the fundamcn ta l th eorem : (x )= y ,, + ('fll )d l -;.o.-.-. ­ ::; - ;.; ~ - Illay bc defined by average VII[lIe = 'h-=- / f(xl dx -Integral curve Imag ine that a run cti on f determines a slope fix) lor eac h x Plac ing linc segments w ith s lo pe fi x) at po ints (x.y) fo r va ri ousy, and doi ng thi s lor va rio us x , one gets a slope field A n antidc ri vati ve o ffi s a fu nc tion whose graph is tange nt to th e slope field at each po int T he gra ph o f the a nt ide ri va ti vc is called a n integ ral curve of the slope fi e ld • Solution to initial value problem The so lution to the diiTerential eq umiony ' ",/"( x) with initia l va luey(xnl = y" is F unda m e nta l T h e orem fl.x) the arca accu lllul ated up to x Iffis negati ve, the integral is the negat ive o f the area • Average value The awragc va llie 011 over a n int erval [a ,h] i for o ne-si de d de rivatives " (l cos ' l dl , si nce ! 1- s in 'l = cos I for () < 1< 1[/ T he for mul a is ollen used in revcrse, staJ1ing w ith ], "1-'(.9 (x ))O' Cddx ec Technic/iII!s on pg • i'atural logarithm A rigorous defini tion is In x = (' Ida T he change of variable for mula with 11 = I I I U , , , I yiel ds / , /., - I /., \ a rill = , I -cc1- d l = - _, I dt sho\\ ing that In ( f Ix) = ~I nx The other elementary propertie, of the natural log can likewise be easily derived from thi, definition in this approach, a n inverse function is deduced a nd is derined to be the natural exponent ial function •, C n INTEGRATION FORMULAS Other routine integration-by-parts integrands arc arcsinx, XCOSX, and xe UX , • Rational functions Every rational function may be written as a polynomial plus a proper rational function (degrec of numerator less than degree of denominator) A proper rational function with real coefficients has a partial fraction decomposition: It can be written as a sum with each summand being either a constant over a power of a lincar polynomial or a linear polynomial over a power ofa quadratic A factor (X-C)k in the denominator of the rational function implies there could be summands Inx xnInx xsinx, • Basic indefinite integrals Each lormula gives just one antidcrivativc (all others dillering by a constant from that given) and is valid on any open interval where the integrand is defined: ~ 1/ ! * J x"dx=~(n n+l f 1dx=lnlxl x -1) fe"xdx=e;x(k * O) feos x dx=sin x fsin x dx= -cns x fl'~:' =arctan x x f~=81'Csin Ii-x' • Further indefinite integrals The above conventions hold: J~()t x dx=ln lsin xl ftan x dx=lnlsec xl A.! +",+~" x-c (x-c)" A factor (x'+bX+C)k (the quadratic not having real roots) in the dcnominator implies there could be summands Isec x dx=lnlsec x + tan x l x'+bx+c " J~sc x dx=ln lcsc x+cot xl Math software can handle the work, but the following case l x-al f~=lln x'-a' 2a x+a Iix ldx= lxlxl X f l x-~-+\: a­.,=lnlx+lx'+a' l=sinh dx=cosh x 1~+lna ,=Inlx+lx'a'icosh 1~+lna fld~ x-a­ (take positive values for cosh· l ) flx' ± a'dx=!xl x'a' ± ~Inlx+ Ix' ± a' i (Take same sign + or - throughout) fla 2 • Common definite integrals: I l' 1" x"dx= -n+l- II fU - Idll,ju-"du(tI > O,fll(U'+ll "du (handled with substitution and fill'+l) "du (handled with w=1I +1) , II = tan t) ,l th~ integrand is ),dx= ) , + < 2.4 GEOMETRY • Areas of plane regions, Consider a plane region admitting an axis slIch that sections perpendicular to the a.xis \ur) in lcngth according to a known function L(p) tl"'I'''' b The area of a strip of width I'!.p perpendicular to Ih~ axis at p i, M = L(pll'!.p, and the total area is IMPROPER INTEGRALS Over [1I.bJ is IIJ on [II,B] for all B>II, then I" j(x)dx ~ ,!i lll " jlx)dx provided the limit exists • Substitution Refers to the Change of variable formula (see the Thmrv section), but ollen the formula is used in reverse For an integral recognized to have the form f."F(y(x»)y' (x)dx (with F and g' continuous), you can put tllI=g'(x)dx, and modify the limits of integration [ "F(,q(xl)y' (x)dx= j~(I("1 F(u)dll ,1/((1) In effect, the integral is over a path on the II-axis traced out by the flJJlction g (I I' g(b) = g(a) [the path returns to its start], then the integral is zero.) E.g., u= I+x' yiclds l_ x _ -1 [I _ l _ ? d -11 (1+ x, ') -l+x'cx- o l+x'_x x- n Substitution may be used lor indefinite integrals E.g l we A= (,L(p )dp E.g" /J iCJ I "j(x)dx = lim 1."j(x)c/x d,1 ;1 • •t In each case, if the limit exists, the improper integral converges and otherwise it diverges For f defined on (_ 00,00) and integrable on every bounded interval, · f ~jlx)dx = ,[i~ l'j(x)dx+ ,li~.,["j(x)dx deI (the choice of c being arbitrary), provided each integral on the right converges • Singular integrands Iff is defined on (lI ,b] but not at x=a and is integrable on closed subintervals of(a,b], then ),"" j(x)dx (hI = }i!~ Ih, j(x)dx provided the limit exists A similar definition holds if the integrand is defined on [1I,b) E.g., l',,4-x' ".!-.,dx lim "- ' is II l',,4-x' ~dx = 1J ,lim -., arcsin(f)=!f, 2 ' , f~ E.g l+x dx-lfdll-llna-lln(l+x') u - -2 ' ' -2 ­ f y(x)",1I (x)dx ,z:tl,I I fY'(x) y(x)dx-ln1y(x)l, "~ _lI(X)" I (C) t • Volumes of solids Consider a solid admitting an axis slich that cross-sections perpendicular to the axis vary in arca according to a known function A(p), a"'p",b The volume of a slab of' thickness I'!.p pl'rpcndi"ular to (l For indefinite integration, fll dv=uu- fv dll, The procedure is used in derivations where the iemetions arc gencral, as well as in explicit integrations You don't need to usc "II" and "v." View the integrand as a product with one factor to be integrated and the other to be differentiated; the integral is the integrated factor times the one to be differentiated, minus the integral of the product of the two new quantities The t~lctor to be integrated may be I (giving v=x) E.g., farctan x dx=x arctan x- f 1':x ,dx ih~ axis at p is III ' =.1(plt!.{', and the total I'olumc is V = "('"i\(p)dp, Lg a pyramid a (' V=j""/I.(z)dz= ("rrrlz)'dz ct The common formula is j'''1t dU=llul" -j'''u dll bp : IIp a ·Solids of re\olution Consider a solid of r~I'()lution det~rmined by a known radius ilmction r(:) , a"' :'" h, along it s axis of revolution The area of th~ cross·,,,ctional "disk" at : is A (:)=nr(:)l and the I'olum e is feu rxly' (x)dx=e yrxl , ['u(x)u' (X)dX=Il(X)U(x)l~ - ['U(X)Il' Cddx ~3"~ V= f~'( l - ~r dz = S~lh , n '/4-x' dx=arcsm 2-2 • Integration by parts Explicitly, [a,b] Sometimcs it i simpler to I' iew a region as bounded by two graphs "over" the y·axis, in which case the integration variabk isy having square horiLollwi Cfllss-scctions with bottom ~ide length !\ and height Jr has cross-st:L:tional area A(:)=[.\'( I-://r)]' at height Ih ,,)!tUllC is thu> Singular Integrand , I"." if/(x) - jlx)idx prm i(kd K(X) "-f(x) on ~O admits all N suth that a,, >Jl h,r all "" N then one writes a,, -+ oo E.g if Irl < I then r" O: if r= I then rll-+I; otherwise r" diverges and ifr> r" -x , • Boun 'd n unotone Sl ph Il ;\11 iilcr~a:-.ing sequence' that is bounded above converges (to OJ limit less thall or equal to any hound) This is a fundamental "let about the real numhcrs, and is basi~ to series convcrgt.:llcc tL·~t S dy rewntten lit =ky suggests y =kdt where lyFkt+c In this way, one finds a solution y=CeAt On any open interval every solution must have that form, because y'=ky implies 1ft (ye M), where yr kl is constant on the interval Thusy=Ce kl (C real) is the general solution The unique solution with y(a)=y" is y=y"ek(l "J The trivial solution isy O solving any IVP y(a)=O • (.eneral first-order line r DE Consider y'+p(t).I' =q(l) The solution to the associated homogeneous equation h'+p(t)1r =0 (dhlh =-p(t)dt) with h(a)=\ is hU)=expl- I'p(u)dul If I' is a solution to the original DE then (ylh)'=qlh where y(t)=Y y=h fq / h The solution - [q(u)h(u) 'dul with y(