- 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...
.. .
