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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 Thursday October 4 httprnathariz0naeducushing250ahtrnl 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 EXAMPLE le JCde y eXZ EXAMPLE le JCde y eXZ decreasing EXAMPLE le JCde dy x2 y 639 LEFTn amp RIGHTn decreasing provide error bounds EXAMPLE le JCde y eXZ concavity changes at x J5 2 x 0707 EXAMPLE le JCde dy x2 y 639 a 2xe concavity Changes at d2 x 2s0707 L2 22x2 1e x dx TRAPn amp MIDn provide no error bounds EXAMPLE 1 2 f 6 dx y eXZ 0 RIGHTn 2 05733 EXAMPLE 1 2 f 6 dx y eXZ 0 RIGHTn LEFTn 2 05733 08894 EXAMPLE 6 dx y eXZ 0 RIGHTn LEFTn 2 05733 08894 4 06640 08220 EXAMPLE 1 x2 6 dx y eXZ 0 n RIGHTn LEFTn 2 05733 08894 4 06640 08220 8 07064 07854 EXAMPLE 1 x2 6 dx y eXZ 0 n RIGHTn LEFTn 2 05733 08894 4 06640 08220 8 07064 07854 16 07268 07663 EXAMPLE 6 dx y eXZ 0 n RIGHTn LEFTn 2 05733 08894 4 06640 08220 8 07064 07854 16 07268 07663 32 07369 07566 x 64 07419 07517 128 07444 07493 256 07456 07481 512 07462 07474 EXAMPLE fol 2 6 dx n TRAPn 2 07314 4 07430 8 07459 16 07466 32 07468 x 64 07468 128 07468 256 07468 512 07468 EXAMPLE fol 2 I 2 1 0 1 2 Decide on stopping rule Number of digits left to right that have stabilized 6 dx n TRAPn 2 07314 4 07430 8 07459 16 07466 32 07468 x 64 07468 128 07468 256 07468 512 07468 EXAMPLE fol 2 6 dx n TRAPn MIDn 2 07314 07546 4 07430 07487 8 07459 07473 16 07466 07469 32 07468 07469 x 64 07468 07468 128 07468 07468 256 07468 07468 512 07468 07468 EXAMPLE 1 2 f e x dx x 07468 0 n TRAPn MIDn 2 07314 07546 4 07430 07487 8 07459 07473 16 07466 07469 32 07468 07469 g x 64 07468 07468 128 07468 07468 256 07468 07468 512 07468 07468 Assuming 07468 is the correct answer note that TRAP amp MID provide more accuracy for each n n RIGHTn LEFTn 2 05733 08894 4 06640 08220 8 07064 07854 16 07268 07663 32 07369 07566 64 07419 07517 128 07444 07493 256 07456 07481 512 07462 07474 n TRAPn MIDn 2 07314 07546 4 07430 07487 8 07459 07473 16 07466 07469 32 07468 07469 64 07468 07468 128 07468 07468 256 07468 07468 512 07468 07468 Assuming 07468 is the correct answer note that TRAP amp MID provide more accuracy for each n TRAP amp MID converge faster as 72 increases n RIGHTn LEFTn 2 05733 08894 4 06640 08220 8 07064 07854 16 07268 07663 32 07369 07566 64 07419 07517 128 07444 07493 256 07456 07481 512 07462 07474 n TRAPn MIDn 2 07314 07546 4 07430 07487 8 07459 07473 16 07466 07469 32 07468 07469 64 07468 07468 128 07468 07468 256 07468 07468 512 07468 07468 Assuming 07468 is the correct answer note that TRAP amp MID provide more accuracy at each step TRAP amp MID converge faster as n increases n LEFTn ERRORn n TRAPn ERRORn 2 08894 1426 x10391 2 07314 1540 x10392 4 08220 7520 x10392 4 07430 3800 x10393 8 07854 3860 x10392 8 07459 9000 x10393 16 07663 1950 x10392 16 07466 2000 x103 32 07566 9800 x10393 32 07468 00000 64 07517 4900 x10393 64 07468 00000 128 07493 2500 x10393 128 07468 00000 256 07481 1300 x10393 256 07468 00000 512 07474 6000 x10393 512 07468 00000 Assuming 07468 is the correct answer note that TRAP amp MID converge faster as n increases TRAP amp MID provide more accuracy at each step n LEFTn ERRORn 2 08894 19095 4 08220 10070 8 07854 5169 16 07663 2611 32 07566 1312 64 07517 06561 128 07493 03348 256 07481 01741 512 07474 00803 n TRAPn ERRORn 2 07314 2062 4 07430 05088 8 07459 01205 16 07466 00268 32 07468 00000 64 07468 00000 128 07468 00000 256 07468 00000 512 07468 00000 Number of steps doubles Error decreases by a factor of 12 n LEFTn ERRORn n TRAPn ERRORn 2 08894 19095 2 07314 2062 4 08220 10070 4 07430 05088 8 07854 5169 8 07459 01205 16 07663 2611 16 07466 00268 32 07566 1312 32 07468 00000 64 07517 06561 64 07468 00000 128 07493 03348 128 07468 00000 256 07481 01741 256 07468 00000 512 07474 00803 512 07468 00000 Number of steps doubles Number of steps doubles Error Error decreases by a factor of 12 decreases by a factor of 14 11 LEFTn ERRORn n TRAPn ERRORn 2 08894 19095 2 07314 2062 4 08220 10070 4 07430 05088 8 07854 5169 8 07459 01205 16 07663 2611 16 07466 00268 32 07566 1312 32 07468 00000 64 07517 06561 64 07468 00000 128 07493 03348 128 07468 00000 256 07481 01741 256 07468 00000 512 07474 00803 512 07468 00000 Step size decreases by 12 Step size decreases by 12 Error decreases by a factor of 12 Error decreases by a factor of 14 n LEFTn ERRORn n TRAPn ERRORn 2 08894 19095 2 07314 2062 4 08220 10070 4 07430 05088 8 07854 5169 8 07459 01205 16 07663 2611 16 07466 00268 32 07566 1312 32 07468 00000 64 07517 06561 64 07468 00000 128 07493 03348 128 07468 00000 256 07481 01741 256 07468 00000 512 07474 00803 512 07468 00000 Put another way Put another way Error ratio E2n z 2 Error ratio E2n z 4 n LEFTn ERRORn n TRAPn ERRORn 2 08894 19095 2 07314 2062 4 08220 10070 4 07430 05088 8 07854 5169 8 07459 01205 16 07663 2611 16 07466 00268 32 07566 1312 32 07468 00000 64 07517 06561 64 07468 00000 128 07493 03348 128 07468 00000 256 07481 01741 256 07468 00000 512 07474 00803 512 07468 00000 Put another way Put another way Error ratio E2n z 2 Error ratio E2n z 4 n LEFTn Error Ratio 11 TRAPn Error Ratio 2 08894 2 07314 4 08220 1896 4 07430 4053 8 07854 1948 8 07459 4222 16 07663 1980 16 07466 4500 32 07566 1990 32 07468 64 07517 2000 64 07468 128 07493 1960 128 07468 256 07481 1923 256 07468 512 07474 2167 512 07468 Step size ln decreases by 12 Step size ln decreases by 12 Error decreases by factor of 12 Error decreases by factor of 14 These observations are not peculiar to this example Step size ln decreases by 12 Step size ln decreases by 12 Error decreases by factor of 12 Error decreases by factor of 14 These observations are not peculiar to this example LEFT amp RIGHT are rst order methods Errorl S constant x l n Step size ln decreases by 12 Step size ln decreases by 12 Error decreases by factor of 12 Error decreases by factor of 14 These observations are not peculiar to this example LEFT amp RIGHT are rst order methods Errorl S constant x l n Errors decrease at roughly the same rate that the step size decreases Step size ln decreases by 12 Step size ln decreases by 12 Error decreases by factor of 12 Error decreases by factor of 14 These observations are not peculiar to this example TRAP amp MID are second order methods 2 lErrorl g constant gtlt I l Errors decrease as roughly the square of the step size decreases LEFT amp RIGHT TRAP amp MID 2 ETTOT S COHSta t Xi lErrorl g constant gtlt n LEFT amp RIGHT TRAP amp MID 2 ETTOT S COHSta t Xi lErrorl g constant gtlt n Increase 11 by a factor of 10 decreases error by a factor of 1 10 LEFT amp RIGHT Error 3 constant x l n Increase 11 by a factor of 10 decreases error by a factor of 1 10 10 times the work gains 1 decimal of accuracy TRAP amp MID lErrorl g constant gtlt n if LEFT amp RIGHT Error 3 constant x l n Increase 11 by a factor of 10 decreases error by a factor of 1 10 10 times the work gains 1 decimal of accuracy TRAP amp MID 2 1 lErrorl g constant gtlt n Increase 11 by a factor of 10 decreases error by a factor of 1100 LEFT amp RIGHT Error 3 constant x l n Increase 11 by a factor of 10 decreases error by a factor of 1 10 10 times the work gains 1 decimal of accuracy TRAP amp MID 2 1 lErrorl g constant gtlt n Increase 11 by a factor of 10 decreases error by a factor of 1100 10 times the work gains 2 decimals of accuracy LEFT amp RIGHT TRAP amp MID 2 iElTOT S COHSta t Xi lErrorl g constant gtlt n A pth order method would gain p decimals of accuracy for 10 times the work I7 1 lErrorl g constant gtlt n SIMPSON S RULE An order 4 method 2 M1Dn TRAPn 3 erpm a weighted average of MID and TRAP SIMPSON S RULE An order 4 method 1 x2 f 6 dx 0 n SHVIPn 2 07469 4 07468 8 07468 16 07468 X 3 32 07469 64 07468 128 07468 256 07468 512 07468 Section 7 Improper Integrals Section 7 Improper Integrals Examples from Probability Theory 1 x2202 e mZ Section 7 Improper Integrals Examples from Probability Theory 1 e xZZG2 0V2 The famous bell curve Describes What s called a normal probability distribution Section 7 Improper Integrals Examples from Probabilitv Theorv 1 e xZZG2 0V2 The famous bell curve Describes What s called a normal probability distribution for a random variable x varying on the infinite interval 00 lt x lt oo Section 7 Improper Integrals Examples from Probabilitv Theorv 1 2 2 probability random variablex f e x 20 dx mZ lies between a and b The famous bell curve Describes What s called a normal probability distribution for a random variable x varying on the infinite interval 00 lt x lt oo Section 7 Improper Integrals Examples from Probability Theory 6 dx UV 27239 is greater than a The famous bell curve 00 1 x2 202 probability random variable x Describes What s called a normal probability distribution for a random variable x varying on the infinite interval 00 lt x lt oo Section 7 Improper Integrals Examples from Probability Theory 1 2 2 probability random variable x e x 20 dx V 27239 is greater than a One kind of improper integral integration over an in nite interval Section 7 Improper Integrals Examples from Probabilitv Theorv l 1 7T xl x This is an example of a betaprobabilily distribution for a random variable x varying on the finite interval 0 lt x lt l 20 15 10 05 I I I I I x 00 02 04 06 08 10 Section 7 Improper Integrals Examples from Probabilitv Theorv l 1 dx probability random variable x 7T 1lxl x This is an example of a betaprobabilily distribution lies between a and b for a random variable x varying on the finite interval 0 lt x lt l 20 15 10 05 I I I I I x 00 02 04 06 08 10 Section 7 Improper Integrals Examples from Probability Theory Ez 1 1 dx probability random variable x lies between 0 and 12 This is an example of a betaprobabilily distribution for a random variable x varying on the finite interval 0 lt x lt l 20 15 10 05 I I I I I x 00 02 04 06 08 10 21 Section 7 Improper Integrals Examples from Probability Theory probability random variable x lies between 0 and 12 Another kind of improper integral integrand becomes infinite at endpoint Section 7 Improper Integrals Text book considers two types A limit of integration is infinite The integrand becomes infinite

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