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# Class Note for MATH 1431 at UH 2

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This 10 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

Example ff 962 dyc J 1 g 2 3 x 122 3 A Upper sum Lower sum 1 9 1 45 LfP 1 15 41 g m 5625 9 1 1 Ufa Z5 4 91 x 12125 3 LfP lt 132dac 33 13 m 8667 g Ufa 1 V 1 X0 x1 X2 3 x4 5 6 7 X8 5P ffIJTAIIJ1 ffIJ AJJ2 ffIJZAfL39n LfP g SP g UfP for all partitions P of a b 2 dz 1 Example f SltPgt flt gtlt gt 1133 12 lt1 m 8563 LfP m 5625 Ufa x 12125 3 1 26 x2d 33 13 m 8667 1 3 3 b LfP g SP fxdx g UfP De nite Integral as the Limit of Riemann Sums Theorem 1 b x dx lllgillrgo fiA1 l fA2 fAn 2 Sections 53 amp 54 De nite Integral and An tiderivative De nite Integral and Antiderivative a H Fx area from a to x and Fx h area from a to x h Therefore Fx h Fx area from x to x h Efx h if h is small and Fx 11 Fx 5 fx 11 11 h X Theorem 2 Let Fm E m dt Then F f for all x in a b Example for r1752 dt 1 Sk t h th h fF dt e c e grap o 0 1 t2 J A RY 1 2 F 12 gt0 F 122 SigHOfFquot 0 0 behavior of graph concave o concave up T down point of inflection 1 1 t2 dt tan 1 5v Note that 0 Fundamental Theorem of Integral Calculus Theorem 3 In general f1dx Fb Fm where is an antiderivative 0f Function Antiderivative xrl xquot r a rational number 7E l r l sinx cos x cos x sin x S602 x tan x sec x tan x sec x 0502 x cot x cscx cot x csc x Example J13 x2 dx Evaluate x2 dx 1 J A y X Area of the shaded region J13x2 dx 23 6 Step 139 Get an antiderivative for fz 12 Fz 713 Step 21 312d1 F3 7 F1 1 3 1 3726 gc 71 7 3 12 gt 0 gt 12 dz area of the shaded region 1 Example W 7W 2 Evaluate sin 1 dm 42 2 2 smz dz ysinx Step 139 Get an antiderivative for m 7 5m Fm 7 7 0053 Step 2 12 Lwzsinmdas 7 mm 7 F77r2 77mm 7 7005mm 0 7r2 sinz y 0 gt sinzdz is not an area 42 Further Properties of Integral Calculus Theorem 4 1 AbfzgzdzAbfzdzzbgzdz 2 Abafzdzaabfzdz 5 Ab dzAcfzdzzbfzdz 4 Iff 29 on m then Ab zwzzabng Examllnles Examp es 5 Evaluate 1 ll 2176z45dz 0 2 4 2 dz 1 I 1 2 3 47E dz 0 7r4 4 secz2 tan175seczdz 0 10 LGCture 17Sections 5154 Integration J iwen He 1 Sections 51 85 52 Area and De nite Integral Area and De nite Integral y A b Area of Q 2 dCL De nite Integral and Lower Upper Sums y y y 9192 93 a x0 x1 x2 x3 xquot b x area of shaded region is a area of shaded region is an lower sum forf upper sum forf Area of Q 2 Area of 21 Area of 22 o o o Area of n LfP mlAcrl mgAcvg o o o mnAacn b LfP g f dCL S UfP for all partitions P of a b

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