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# Class Note for MATH 250A at UA

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Date Created: 02/06/15
MATH 250a Fall Semester 2007 Section 2 J M Cushing Tuesday October 2 httpmathariz0naeducushing250ahtml Chapter 7 Integration Sections 1 6 Two main themes gt Calculation of antiderivatives integrals One goal is to evaluate integrals using the Fundamental Theorem 0f Calculus gt Numerical approximation of integrals Chapter 7 Integration Sections 1 6 Two main themes gt Calculation of antiderivatives integrals One goal is to evaluate integrals using the Fundamental Theorem 0f Calculus gt Numerical approximation of integrals An antiderivative is not available 1 x2 f e dx 0 Chapter 7 Integration Sections 1 6 Two main themes gt Calculation of antiderivatives integrals One goal is to evaluate integrals using the Fundamental Theorem 0f Calculus gt Numerical approximation of integrals An antiderivative is not available A formula for the integrand is not available 5 Numerical approximation of integrals The accumulated amount of a chemical drug over time is important in toxicity studies Cumulative effect dose of a drug amount X length of time present 5 Numerical approximation of integrals The accumulated amount of a chemical drug over time is important in toxicity studies Cumulative effect dose of a drug amount X length of time present OK if the amount remains constant over the time interval 5 Numerical approximation of integrals The accumulated amount of a chemical drug over time is important in toxicity studies Cumulative effect dose of a drug amount X length of time present Otherwise we integrate amount of drug AU over total time of interest b Cumulative dose 2 Ia Atdt 5 Numerical approximation of integrals Estimate the cumulative dose from the data below from hour a to b b j Atdt a Timer hour 0 l 2 3 4 5 6 7 8 9 10 AmountA 005 046 087 054 043 036 028 021 016 012 009 mgml 5 Numerical approximation of integralg Data 10 O 08 a gas a o 04 o o 0 V O 02 O O o O I I I I I I I I U 1 2 3 4 5 6 7 8 10 Time Tlmel 0 1 2 3 4 5 6 7 8 9 10 hour AmountA 005 046 087 054 043 036 028 021 016 012 009 mgml 5 Numerical approximation of integralg Data 10 0 increases a 398 39 in the beginning w o A o o 0 V O 02 O O o O 00 I u I I I I l 0 1 2 3 4 5 6 7 8 10 Time Tlmel o 1 2 3 4 5 6 7 8 9 10 hour AmountA 005 046 087 054 043 036 028 021 016 012 009 mgm1 5 Numerical approximation of integralg Data 10 0 decreases E at the end gas 5 O 04 o o 0 V O 02 O O o O 0390 I I I I I I I I 0 1 2 3 4 5 6 7 8 10 Time Tlmel 0 1 2 3 4 5 6 7 8 9 10 hour AmountA 005 046 087 054 043 036 028 021 016 012 009 mgml 5 Numerical approximation of integralg Data 10 0 also appears E 0398 concave up gas 5 O 04 o o 0 V O 02 O O o O 0390 I I I I I I I I 0 1 2 3 4 5 6 7 8 10 Time Tlmel 0 1 2 3 4 5 6 7 8 9 10 hour AmountA 005 046 087 054 043 036 028 021 016 012 009 mgml 5 Numerical approximation of integrals Definition of the definite integral text page 252 b quot Ia f0 dr quot1330 grew att0 ltt1ltolttnb At ti tH t11ltcllttl 5 Numerical approximation of integrals De nition of the de nite integral text page 252 b a 1 0 d n13 gm Atz at0 ltt1lt lttn b All tl tH 1H ltcl lt11 f t t a 755 Ci ti1 b Figure 523 A general Riemann sum39apprpxima ng f Lit 5 Numerical approximation of integrals De nition of the de nite integral text page 252 b a 1 0 d n13 gm Atz atoltt1lt lttnb mgr 111 tHltcllttl t Figure 523 A general Riemann sum39approxima ng f t dt 5 Numerical approximation of integrals I fldt lim Righthand sum 2 lim iftiAl ngtw 171 ngtoo dark blue rectangles 5 Numerical approximation of integrals I fldt lim Righthand sum lim ZnftiAl ngt00 I l gt00 i1 I f 1 dz lim Lefthand sum 2 quot1 f liAz ngt 00 light blue rectangles 5 Numerical approximation of integrals n71 I fz dz z 2 fzAz Righthand sum 2 RIGHTn i0 7171 I f 1 dz z 2 f IiAI Lefthand sum LEFTn i0 5 Numerical approximation of integrals In this gure we see that the left rectangles over estimate because f I is decreasing 5 Numerical approximation of integrals In this gure we see that the rihl rectangles under estimate Z is decreasing 5 Numerical approximation of integrals If f t is decreasing then RIGHTn S I f l dz 3 LEFTn 5 Numerical approximation of integrals If f t is decreasing then RIGHTn S I f t dt S LEFTn If f t is increasing then LEFTn S I f t dt S RIGHTn Error Bounds Estimate the cumulative dose during the last 5 hours 10 j At dt z 5 Time I hour O 1 2 3 4 5 6 7 8 9 10 Amount A 005 046 087 054 043 036 028 021 016 012 009 mgml Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 Timer hour 0 l 2 3 4 5 6 7 8 9 10 Amount1 005 046 087 054 043 036 028 021 016 012 009 mgml The data provides the values of Al at endpoints of unit length subintervals so we can calculate LEFT5 and RIGHT5 with At l Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 Timer hour 0 l 2 3 4 5 6 7 8 9 10 Amount1 005 046 087 054 043 036 028 021 016 012 009 mgml The data provides the values of Al at endpoints of unit length subintervals so we can calculate LEFT5 and RIGHT5 with At 1 Note datal taken at hours 5 6 7 8 9 10 decreases Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 Timer hour 0 1 2 3 4 5 6 7 8 9 10 Amount1 005 046 087 054 043 036 028 021 016 012 009 mgml If we assume Al decreases during the last ve hours then RIGHT5 s 510 Atdt s LEFT5 Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 I T112165 5 6 7 8 9 10 Amount1 036 028 021 016 012 009 mgml I I RIGHT5 s go1mg s LEFT5 Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 I T112165 5 6 7 8 9 10 Amount1 036 028 021 016 012 009 mgml I I RIGHT5 s go1mg s LEFT5 s go10W s LEFT10 Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 I T112165 5 6 7 8 9 10 Amount1 0361028 021 016 012 009 mgml I I RIGHT5 s go1mg s LEFT5 O28O21O16O12009 s go10W s LEFT10 Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 I T112165 5 6 7 8 9 10 Amount1 036 028 021 016 012 009 mgml I I RIGHT5 s go1mg s LEFT5 086 s I510Atdt s LEFT10 Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 I T112165 5 6 7 8 9 10 Amount1 036 028 021 016 012 009 mgml I I 1 RIGHT5 s go1mg s LEFT5 086 s E04000 s O36O28O21O16O12 Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 I T112165 5 6 7 8 9 10 Amount1 036 028 021 016 012 009 mgml I I RIGHT5 s go1mg s LEFT5 08645104000913 Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 I T112165 5 6 7 8 9 10 AmountA 036 028 021 016 012 009 mgml I I RIGHT5 s go1mg s LEFT5 08645104000913 How might we improve these approximations Estimate the cumulative dose during the last 5 hours 10 I Atdt 5 Timer I I I 5 6 7 8 9 10 hour AmountA 036 028 021 016 012 009 mgml I I I I In using LEFT5 we effectively assumed the dose was constant at 036 from hour 5 to 6 Estimate the cumulative dose during the last 5 hours 10 I Atdt 5 Timer I I I 5 6 7 8 9 10 hour AmountA 036 028 021 016 012 009 mgml I I I I In using LEFT5 we effectively assumed the dose was constant at 036 from hour 5 to 6 In using RIGHT5 we effectively assumed the dose was constant at 028 from hour 5 to 6 Almost assuredly neither assumption is true Estimate the cumulative dose during the last 5 hours 10 I Ata t 5 Timer I I I 5 6 7 8 9 10 hour AmountA 036 028 021 016 012 009 mgml I I I I In using LEFT5 we effectively assumed the dose was constant at 036 from hour 5 to 6 In using RIGHT5 we effectively assumed the dose was constant at 028 from hour 5 to 6 Almost assuredly neither assumption is true Most likely the dose continuously decreased from 036 to 028 from hour 5 to 6 Estimate the cumulative dose during the last 5 hours 10 I Atdt 5 Timer I I I 5 6 7 8 9 10 hour A tA mun 036 028 021 016 012 009 mgml I I I I Perhaps we can obtain an improved approximation by using the average of the sample doses Estimate the cumulative dose during the last 5 hours 10 I Atdt 5 Timer I I I 5 6 7 8 9 10 hour AmountA 036 028 021 016 012 009 mgml I I I I Perhaps we can obtain an improved approximation by using the average of the sample doses For example assume a dose of 032 om hour 5 to 6 Estimate the cumulative dose during the last 5 hours 10 j At d z 5 Time I I I I hour 5 6 7 8 9 10 Amount A 036 028 021 016 012 009 mgml I I l l Perhaps we can obtain an improved approximation by using the average of the sample doses 036 028 For example assume a dose of f 032 om hour 5 to 6 Mathematically we would then be using a rectangle approximation whose height the average of the endpoint heights Estimate the cumulative dose during the last 5 hours 10 j At d z 5 Timer I I I 5 6 7 8 9 10 hour Amount A 036 028 021 016 012 009 mgml I I l l Perhaps we can obtain an improved approximation by using the average of the sample doses 036 028 For example assume a dose of f 032 om hour 5 to 6 Mathematically we would then be using a rectangle approximation whose height the average of the endpoint heights ie the heights used in LEFT5 and RIGHT5 LEFTn RIGHTn 2 Trapezoid Rule Define TRAPn LEFTquot RIGHT 2 Tra ezoid Rule LEFTn RIGHTn 2 Why is this called the Trapezoid Rule TRAPn Tra ezoid Rule LEFTn RIGHTn 2 Why is this called the Trapezoid Rule TRAPn area of a trapezoid 2 base X average height Tra ezoid Rule LEFTn RIGHTn 2 Why is this called the Trapezoid Rule TRAPn area of a trapezoid 2 base X average height hh hl b122 172 Trapezoid Rule LEFTn RIGHTn TRAPn 2 f concave down Trapezoid underestimates Tra ezoid Rule T RIGHT TRAN LEF n 2 n f concava down f concave up Trapezoid underestimates Trapezoid overes rnales Trapezoid Rule LEFT RIGHT n TRAPn 2 j concave down Trapezoid underestimates Trapezoid eres1imales Iffl is concave up then I fl dz S TRAPn Trapezoid Rule LEFTn RIGHTn 2 TRAPn j concave down Trapezoid underestimates f concave up Trapezoid overestimates If f I is concave up then 1 dz S TRAPn b If f t is concave down then TRAPO39I g Ia f t dt Trapezoid Rule LEFT RIGHT TRAPn 2 n j concave down f concave up Trapezoid underestimates Trapezoid overestimates Iffl is concave up then I fl dz S TRAPn b If f t is concave down then TRAPn g Ia f t dt Onesided error bounds only Trapezoid Rule We can get twosided error bounds by using another approximation method one that uses averages on the horizontal axis instead of the vertical axis Midpoint Rule jfrdrf tit 2 139 At MIDn midpoints 0f the subintervals Midpoint Rule Iftdtzf t1quot 24 At MIDn 139 i H Midpoint Rule I ft dt z ffquot 2ti1jm MIDn 139 i H ti 51 Midpoint Rule I ft dt z ffquot 2ti1jm MIDn 139 i H ti 51 Midpoint Rule Iftdtzf t1quot 24 At MIDn 139 i H ti 51 Midpoint Rule I ft dt z ffquot 2ti1jm MIDn Rotate the top around the midpoint 139 i H ti 51 Midpoint Rule I ft dt z ffquot 2ti1jm MIDn Rotate the top around the midpoint Congruent triangles have equal areas Midpoint Rule I ft dt z ffquot 2ti1jm MIDn The created trapezoid has Rotate thet0 the same area as the around the mldpolnt original rectangle Congruent triangles have equal areas 139 i l l ti ti1 Midpoint Rule I f t dt z f 4 24 At MIDn The created trapezoid has ROtate thet0p the same area as the around the mldpomt original rectangle Congruent triangles 110 matter what the have equal areas rotation angle is ti ti 1 E 21 Midpoint Rule I ft dt z ffquot 2ti1jm MIDn Rotate the top so as to become tangent to the curve ti tiH ti ti1 Midpoint Rule I ft dt z ffquot 2ti1jm MIDn Rotate the top so as to become tangent to the curve ti ti 1 ti ti1 Midpoint Rule I ft dt z ffquot 2ti1jm MIDn The area of tangential trapezoid equals the area of the midpoint rectangle ti ti 1 ti ti1 Midpoint Rule Iftdtzf I I 1211jm MIDn Midpoint Rule MIDn 139 i H ti 51 Midpoint Rule 5mm z ffquot ljm MIDn 2 Midpoint Rule MIDn Trapezoid rule TRAPn t t r 2H 1 Midpoint Rule I ft dt z ffquot 2ti1jm MIDn Midpoint Rule MIDn tangent trapezoid Trapezoid rule TRAPn secant trapezoid 139 i H ti 51 Midpoint Rule If f t is concave down then TRAPn S I f t dt S MIDn Midpoint Rule MIDn tangent trapezoid Trapezoid rule TRAPn secant trapezoid 139 i H ti 51 Midpoint amp Trapezoid Rules b If f t is concave down then TRAPn S J a f t dt S MIDn b If f t is concave up then MIDn S J a f t dt S TRAPn Error Bounds Estimate the cumulative dose during the last 5 hours 10 I Atdt 5 Timer I I I I 5 6 7 8 9 10 hour AmountA 036 028 021 016 012 009 mgml I I I I We can use the Trapezoid Rule it uses only endpoint data Estimate the cumulative dose during the last 5 hours 10 I Atdt 5 Timer I I I I 5 6 7 8 9 10 hour AmountA 036 028 021 016 012 009 mgml I I I I We can use the Trapezoid Rule it uses only endpoint data but we can t use the Midpoint Rule no data at midpoints Estimate the cumulative dose during the last 5 hours 10 I Atdt 5 Timer I I I I 5 6 7 8 9 10 hour AmountA 036 028 021 016 012 009 mgml I I I I Recall that assuming A is decreasing RIGHT5 s 0 Atdt s LEFT5 086sjleArdrs113 Estimate the cumulative dose during the last 5 hours 10 j Atdt z 5 Time I I I I I 5 6 7 8 9 10 hour Amount1 036 028 021 016 012 009 mgml I I I I Recall that assuming A is decreasing TRAP5 086 l l3 2 O 995 10 2 RIGHT5 s 5 Atdt s LEFT5 average 086sjleArdrs113 Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 I T1233 5 6 7 8 9 10 AmountA 036 028 021 016 012 009 mgml Recall TRAP52086ll32039995 RIGHT5 s 0 Atdt s LEFT5 086sjleArdrs113 If we assume Al is concave up then Smlam g 0995 Estimate the cumulative dose during the last 5 hours 10 j Atdt 5 Timer I I I I 5 6 7 8 9 10 hour AmountA 036 028 021 016 012 009 mgml I Recall 086ll3 TRAP5 T 0995 RIGHT5 s 0 Atdt s LEFT5 If we assume Al is concave up then 10 10 0865 Atdt 113 5 Atdtso995 08610Ardrso995 Data 10 j Atdt o 0 08 E goe E n 04 0 O V O 02 o o O 0390 I I I I I I I 1 2 4 6 8 10 Time Timer hour 0 1 2 3 4 5 6 7 8 9 10 Arno ntA u 005 046 087 054 043 036 028 021 016 012 009 mgml Data 10 j Atdt z o 0 08 E 06 We can use rectangle amp trapezoid rules 5 f M o 0 but the error bounds don t apply V O 02 O o O 0390 I I I I I I I 1 2 4 6 8 10 Time Time I hour 0 1 2 3 4 5 6 7 8 9 10 Arno ntA u 005 046 087 054 043 036 028 021 016 012 009 mgml Data 10 j Atdt o 0 LEFT10 E lt11 n 04 0 O V O 02 o 00 o 0 Time Timer hour 0 1 2 3 6 7 8 10 Amount1 005 046 087 054 028 021 016 009 mgml Data 10 j Atdt z o 0 LEFT10 348 E T 04 o o V O 02 o 00 o 0 Time Time I hour 0 1 2 3 6 7 8 10 Amount1 005 046 087 054 028 021 016 009 mgml Data 10 j Atdt o o LEFT10 348 E o RIGHT10352 V O 02 o o O 0 1 2 4Time6 8 10 Timer hour 0 1 2 3 4 5 6 7 8 9 10 Amount1 005 046 087 054 043 036 028 021 016 012 009 mgml t t 1 Data 10 j Atdt o 0 LEFT10 348 a o RIGHT10352 V 02 o o 0 1 2 4Time6 8 10 Timer hour 0 1 2 3 4 5 6 7 8 9 10 Amount1 005 046 087 054 043 036 028 021 016 012 009 mgml Amount P a P 33 03 no G A P N 00 Data Time ISO1mm z LEFT10 348 RIGHT10 352 TRAP10 350 Because the data lacks monotonicity andor concavity uniformly across the interval the theoretical error bounds do not apply We can t relate any of these approximations to the exact value of the integral

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