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# Class Note for MATH 1431 with Professor Caboussat at UH

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This 23 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 15 views.

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Date Created: 02/06/15

MATH 1431 19328 Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 0 On the last day to drop a course or withdraw with a 39W39 the front desk of the Math Department may sign drop forms ifl am not available httpwwwmathuheduNcaboussatmath1431 l Chap 5 Upper and Lower Riemann Sums 132 2 a uuuu mm Luwu mm 1 m Partition of 37 b a X07X1 7Xr b Definition Upper sum UfP Mlel MQAXQ MnAXn Lower Sum LfP mlel 772sz mnAxn METH 1411 Definition A function f defined on an interval 3 b is integrable on 3 b if there is one and only one number I that satisfies the inequality LfP I UfP for all partitions P of 3 b The unique number I is called the definite integral or more simply the integral of f from 3 to b and is denoted by I fxdx Notations abfxdxor abftdtor abfsds Example We Have Seen More When F is non negative the area under the curve fx between a and b is given by MW HWHamp where FX is an anti derivative of f Therefore METH 1411 b a fxdx Fb 7 Fa whether f is non negative of not 0 When f is nonnegative the integral represents the area between the graph of f and the X aXis 0 When f is not nonnegative what is the signification of the integral METH 1411 Find the area bounded by the curve fx sinx and the X axis for 0 g X g 7139 AX sinx AX 7cosx C AO 0 implies C 1 and AX 17 cosx Total Area A7r 17 cos7r1771 2 Compute the lower and upper Riemann sums for fx sinx over 077139 with respect to the partition P 07r2737r477r METH 1411 Compute the lower and upper Riemann sums for fx sinx over 077139 with respect to the partition P 07r2737r477r METH 1411 1 Compute X3X2X1dx 0 5 Compute dx 0 53 The Fynction Fx ftdt a Find the area bounded by the curve fx X2 7 3X 7 4 and the X aXis on the interval 727 3 METH 1411 Find the area bounded by the curve fx X2 7 3X 7 4 and the X aXis on the interval 727 3 METH 1411 71 Compute X2 7 3X 7 4 dx 72 3 Compute X2 7 3X 7 4 dx 71 3 Compute X2 7 3X 7 4 dx 72 rmeH 1411 If f is continuous on 37 b and a lt c lt b then ftdtbetdtAbftdt Area Area Total Area J wt TH 1411 1 When a lt b Consequence HWTH 1431 1 Examples 0 Compute X2 X dx 1 0 Compute cosxdx So far we know abftdt Remember In Chapter 3 we defined Fl before defining Fx as a function of X Question Can we define the integral as a function of X The answer is yes By definition Fx ftdt VHWTH 1431 1 In Terms of Surface rmeH 1411 Let f be a continuous functions on 37 b The function F defined by X Fx ftdt is continuous on 37 b differentiable on 37 b and has derivative FX fx7 for all X 6 37 b METH 1411 Examples On the interval 01 If ft 1 Fx If ft t2 t1 Fx If ft sin5t Fx rmeH 1411 X 7r Compute tzisint2 dx and tzisint2dt 0 0 rmeH 1411

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