1.Section5.3 EvaluatingDeniteIntegrals V63.0121, CalculusIApril21, 2009 AnnouncementsFinalExamisFriday, May8, 2:003:50pmFinaliscumulative; topicswillberepresentedroughlyaccordingtotimespentonthem . Imagecredit: docman . . . . .. .

2. Outline . . . . . . 3. Thedeniteintegralasalimit DenitionIf f isafunctiondenedon [a, b], the deniteintegralof f from ato b isthenumberb n f(x) dx = lim f(ci ) x n a i=1ba , andforeach i, xi = a + ix, and ci isapointwhere x =nin [xi1 , xi ].... .. . 4. Notation/Terminology bf(x) dxa integralsign (swoopy S) f(x) integrand a and b limitsofintegration (a isthe lowerlimit and b the upperlimit) dx ??? (aparenthesis? aninnitesimal? avariable?) Theprocessofcomputinganintegraliscalled integration. . . . . . 5. PropertiesoftheintegralTheorem(AdditivePropertiesoftheIntegral)Let f and g beintegrablefunctionson [a, b] and c aconstant.Thenb c dx = c(b a) 1.ab b b[f(x) + g(x)] dx =f(x) dx + g(x) dx. 2.a a abbcf(x) dx = c f(x) dx. 3.aab b b[f(x) g(x)] dx =f(x) dx g(x) dx. 4.a a a . . . . . . 6. MorePropertiesoftheIntegral Conventions: ab f(x) dx = f(x) dx ba af(x) dx = 0aThisallowsustohave cb c f(x) dx =f(x) dx + f(x) dx forall a, b, and c. 5.aab .. . . . . 7. ComparisonPropertiesoftheIntegralTheoremLet f and g beintegrablefunctionson [a, b]. 6. If f(x) 0 forall x in [a, b], then b f(x) dx 0 a7. If f(x) g(x) forall x in [a, b], then bb f(x) dx g(x) dx aa8. If m f(x) M forall x in [a, b], then b m(b a) f(x) dx M(b a) a . . . . . . 8. Outline . . . . . . 9. Socraticproof Thedeniteintegralofvelocitymeasuresdisplacement(netdistance)ThederivativeofdisplacementisvelocitySowecancomputedisplacementwiththeantiderivativeofvelocity? . . . . . . 10. TheoremoftheDayTheorem(TheSecondFundamentalTheoremofCalculus) Suppose f isintegrableon [a, b] and f = F foranotherfunction F, then b f(x) dx = F(b) F(a). a .. . . . . 11. TheoremoftheDayTheorem(TheSecondFundamentalTheoremofCalculus) Suppose f isintegrableon [a, b] and f = F foranotherfunction F, then b f(x) dx = F(b) F(a). a Note InSection5.3, thistheoremiscalledTheEvaluationTheorem. Nobodyelseintheworldcallsitthat. .. . . . . 12. Proving2FTCba Divideup [a, b] into n piecesofequalwidth x = as n usual. Foreach i, F iscontinuouson [xi1 , xi ] anddifferentiable on (xi1 , xi ). Sothereisapoint ci in (xi1 , xi ) with F(xi ) F(xi1 )= F (ci ) = f(ci ) xi xi1Or f(ci )x = F(xi ) F(xi1 ). . . . . . 13. Wehaveforeach if(ci )x = F(xi ) F(xi1 )FormtheRiemannSum:nn (F(xi ) F(xi1 )) Sn = f(ci )x =i=1i=1 = (F(x1 ) F(x0 )) + (F(x2 ) F(x1 )) + (F(x3 ) F(x2 )) + + (F(xn1 ) F(xn2 )) + (F(xn ) F(xn1 )) = F(xn ) F(x0 ) = F(b) F(a) . . . . . . 14. Wehaveshownforeach n, Sn = F(b) F(a)sointhelimitbf(x) dx = lim Sn = lim (F(b) F(a)) = F(b) F(a)nna .. .. . . 15. Example Findtheareabetween y = x3 andthe x-axis, between x = 0 and x = 1. . . . . . . . 16. ExampleFindtheareabetween y = x3 andthe x-axis, between x = 0 andx = 1.Solution 11x4 1x3 dx =A= =44.00. . . . . . 17. ExampleFindtheareabetween y = x3 andthe x-axis, between x = 0 andx = 1.Solution 11x4 1x3 dx =A= =44.00Hereweusethenotation F(x)|b or [F(x)]b tomean F(b) F(a).aa. . . . . . 18. Example Findtheareaenclosedbytheparabola y = x2 and y = 1.. ... . . 19. Example Findtheareaenclosedbytheparabola y = x2 and y = 1. .. ... . . 20. Example Findtheareaenclosedbytheparabola y = x2 and y = 1. .Solution[ ]1 [ ()] 1x31 142 A=2x dx = 2 =2 =33 3311 . . . . . . 21. Outline . . . . . . 22. TheIntegralasTotalChangeAnotherwaytostatethistheoremis: bF (x) dx = F(b) F(a),a or theintegralofaderivativealonganintervalisthetotalchangebetweenthesidesofthatinterval. Thishasmanyramications:.. . . . . 23. TheIntegralasTotalChangeAnotherwaytostatethistheoremis: b F (x) dx = F(b) F(a),a or theintegralofaderivativealonganintervalisthetotalchangebetweenthesidesofthatinterval. Thishasmanyramications: TheoremIf v(t) representsthevelocityofaparticlemovingrectilinearly,then t1 v(t) dt = s(t1 ) s(t0 ). t0 .. . . . . 24. TheIntegralasTotalChangeAnotherwaytostatethistheoremis: bF (x) dx = F(b) F(a),a or theintegralofaderivativealonganintervalisthetotalchangebetweenthesidesofthatinterval. Thishasmanyramications: TheoremIf MC(x) representsthemarginalcostofmaking x unitsofaproduct, then xC(x) = C(0) +MC(q) dq. 0 .. . . . . 25. TheIntegralasTotalChangeAnotherwaytostatethistheoremis: bF (x) dx = F(b) F(a),a or theintegralofaderivativealonganintervalisthetotalchangebetweenthesidesofthatinterval. Thishasmanyramications: TheoremIf (x) representsthedensityofathinrodatadistanceof x fromitsend, thenthemassoftherodupto x is x m(x) = (s) ds. 0 .. . . . . 26. Outline . . . . . . 27. A newnotationforantiderivatives Toemphasizetherelationshipbetweenantidifferentiationandintegration, weusethe indeniteintegral notation f(x) dx foranyfunctionwhosederivativeis f(x). . . . .. . 28. A newnotationforantiderivatives Toemphasizetherelationshipbetweenantidifferentiationandintegration, weusethe indeniteintegral notation f(x) dx foranyfunctionwhosederivativeis f(x). Thus x2 dx = 1 x3 + C.3 .. . . . . 29. Myrsttableofintegrals [f(x) + g(x)] dx = f(x) dx + g(x) dxxn+1 xn dx = cf(x) dx = c f(x) dx + C (n = 1) n+11 ex dx = ex + C dx = ln |x| + Cxaxax dx = +C sin x dx = cos x + C ln a csc2 x dx = cot x + C cos x dx = sin x + C sec2 x dx = tan x + C csc x cot x dx = csc x + C 1dx = arcsin x + C sec x tan x dx = sec x + C1 x2 1dx = arctan x + C1 + x2 . ..... 30. Outline . . . . . . 31. Example Findtheareabetweenthegraphof y = (x 1)(x 2), the x-axis, andtheverticallines x = 0 and x = 3. .. .. .. 32. Example Findtheareabetweenthegraphof y = (x 1)(x 2), the x-axis, andtheverticallines x = 0 and x = 3.Solution 3(x 1)(x 2) dx. Noticetheintegrandispositiveon Consider0 [0, 1) and (2, 3], andnegativeon (1, 2). Ifwewanttheareaof theregion, wehavetodo 123(x 1)(x 2) dx (x 1)(x 2) dx +(x 1)(x 2) dx A=0 1 2 [1]1[1 3 ]2 [1 ]3 x3 3 x2 + 2x 0 3 x2 + 2x3323 x 2 x + 2x3x = + 3 (2)21 251511 = += .6666 .. .... 33. Graphfrompreviousexample y . . . . . x . 2 . 3 . 1 . . . . . . . 34. SummaryintegralscanbecomputedwithantidifferentiationintegralofinstantaneousrateofchangeistotalnetchangeThesecondFunamentalTheoremofCalculusrequirestheMeanValueTheorem... .. .