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# Class Note for MATH 1314 with Professor Gross at UH

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COURSE
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PAGES
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KARMA
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This 6 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 13 views.

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Date Created: 02/06/15
M1314 Lessun 18 1 Mach 1314 Lesmn 18 Ana and me In nite 1mm We arenuwreadytu tackle the secundbasxc questmn ufcalculus Aha area quesuun We eah easrly cumpute the zreaunderthe gaph efafuheheh su lung asthe shape quhe regmn eehferrhs m sumethmg fur whreh wehave afurmulafur geumetry Example 1 Suppuse m 5 Fmdthe areaunder the graph hf m hem x I U x 4 Apprmdmz ngArm Under a Curve Nuw Suppuse the areaunda39 the curve is he sumethmg whuse area eah he easrly eempmea We39ll need m develup amethud fur ehmhg sueh an area Example 2 Here we39ll draw seme rectangles m apprehrhare the areaunda39 the curve We eah ndthe areaefeaeh rectangle then add up the areas m appmxlmate the area underthe curve I I Z Eachrectzngle has awxdth hf Ax Thehaghtxs deter mmEd hy Lhevalue af x The area is appmxlmated hy the sum ufthe areas quhe rectangles M1314 LessmlE 2 Eurup 3 Nzxt we39llmcruse than an ufxecunges Whatyau shnuldsee 15 um um number afxecungesmcxuses the areawe campu39z usng 39nsme39hadbecamesmme annual Th An Undnlhz anhufz Fm n Ln be amngmve cman nmanan b Thzn39hearuuf39hexegmmdex m gaph uffxsgvznby A ygLX VM XnAX whzxe x xi xn are mum pm m the mural 11 5 Dream wxdlh Ax The sums arms ufxectmges are calledR39Imann sums undue named mg acmm mathzmmum M1314 Lesson 18 3 Example 5 Use left endpoints and 4 subdivisions of the interval to approximate the area under fx 2x2 1 on the interval 0 2 Example 6 Use right endpoints and 4 subdivisions of the interval to approximate the area under f x 2x2 1 on the interval 0 2 M1314 Lesson 18 4 Example 7 Use midpoints and 4 subdivisions of the interval to approximate the area under fx 2x2 1 on the interval 0 2 Example 8 Suppose f x 1 3x Approximate the area under the graph of f on the interval 0 12 using 6 subdivisions and left endpoints M1314 Lesson 18 5 The De nite Integral Letfbe de ned on 11 b If limfx1 fx2 fxH Axexists for all choices of b a representative points in the n subintervals of 11 b of equal width Ax 7 then this n limit is called the de nite integral of f from a to b The de nite integral is noted by rfxdx limfx1 fx2 fxn Ax The number a is called the lower limit of integration and the number b is called the upper limit of integration A function is said to be integrable on 11 b if it is continuous on the interval 11 b The de nite integral of a nonnegative function fxdx is equal to the geometric area u n m M1314 Lesson 18 6 The definite integral of a general function Page 976 in your book The part of the graph of f that goes below the xaxis is considered negative area Using definite integration we will look how to integrate this function form 0 to 5 ifxdx 7 ifxdx 0 2 From this section you should be able to Explain the procedure used to approximate area under a curve Use Riemann sums to approximate the area under a curve using right endpoinm left endpoints or midpoinm Explain what we mean by definite integral of a nonnegative function or a general function

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