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# Survey of Calculus MATH 160

BSU

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This 4 page Class Notes was uploaded by Breanne Schaden PhD on Saturday October 3, 2015. The Class Notes belongs to MATH 160 at Boise State University taught by Tommy Conklin Jr in Fall. Since its upload, it has received 6 views. For similar materials see /class/218001/math-160-boise-state-university in Mathematics (M) at Boise State University.

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Date Created: 10/03/15

14 THE DEFINITE INTEGRAL We have just seen in the previous chapter that we can calculate left and right hand sums which approximate the signed area under a curve The definite integral is defined as the limit of the lefthand or righthand sum as the number of partitions n goes to infinity Thus each definite integral is a specific real number and the Twil calculate this value Well almost it calculates an approximation that is generally reliable The definite integral is evaluated as a number but we will see that we can also define its upper limit as a variable and thus create a new function The definite integral from a graph CALC 7 Ifxdx By displaying a graph and using the CALC menu we can find the definite integral of a function and see its graphic representation Let39s start with y 25inxand graph it in the ZTrig window Press 2ndCALC to see the integral option 7 Ifxdx You are prompted to set the lower limit and then the upper limit These limits are set in the same way that you have already set bounds using 4 Maximum Remember that they must be within XMax and XMin v1zxinm ME va ue A 2 zero 3m1nmun 4max1num Silntersect 6d3dx xo vo If xgtdx v1zxinoo v1zxinoo VVV Lower Limit HDNV Limit X0 V0 il3139l152 V11755E 395 lfxdx39 Figure 14 I The value of a de nite integral found and shown as shaded area on a graph This number can be interpreted as a measure of shaded area under the function39s graph It may surprise you that the result is an integer but notice that Y1 is twice the sine function and recall from the last chapter that the left and right hand sums ofthe sine function converge to 2 over this interval Tip If there is more than one function graphed then you must select the desired function before using the 7 If x dx command The definite integral as a number MATH fnlnt From the home screen the TI command for the definite integral is in the MATH menu 9 fnlntlt from 39function integral39 which has the general format fnlntfunction variable lower upper The translation from the mathematical symbols of the previous example to T commands is 125inxdx 3 fnlnt25inx X 0 7r Facts about the definite integral Four definite integral facts will be illustrated using simple functions and windows as in Figure 141 You are encouraged to change the function and window to make it more exciting In each of the following examples we show the result graphically and then in the final frame we show the numerical rendition ofthe same result on the home screen It is important that you feel comfortable using both methods of finding the definite integral Tip After a graph with shading has been displayed it is usually desirable to clear the screen before the next graphing This is done by pressing 2ndDRAW 1CIrDraw Using a ZOOM menu selection also gives a fresh graph This is handy when your window is ZStandard ZDecimaI ZTrig or ZPrevious Reversed limit integrals are the negative of one another Unlike most settings where error messages are given whenever XMin gt XMax gt Left Bound gt Right Bound the Upper Limit and Lower Limit can be in either order The result of an order reversal is a sign change ofthe value V1lenll V1innll ntntltV1 1 X 0 7 A f nIntV1Xym04 V V VIin IlntrLimiK 139015527 V117EE 395 ll0 Y0 lttxaxquot1 Figure 142 Limit reversal changes the Sign ofthe de nite integral The intermediate stopover privilege The definite integral can be calculated as a whole from the lower to upper limit or it can be calculated in contiguous pieces This can be thought of as a plane fare where the charge is the same whether you fly nonstop or have an intermediate landing Figure 143 shows that we get the same answer dividing our example function over two particular subintervals V1innll Y1innll J 39 39v L39m39t ig nm39h vo gzii39ra39os39n v1ai77591 IKXMXL23H5331 V1innll V1innll 39 39I up rL39m39t Lg ifgi39i gitz raj 77591 vii13527 v1175 5 lxdx27653665 l nIntWi xwm f nIntV1X011 780972fn1ntlt r 1 X11786972yn4 Figure 143 The de nite integral found in two pieces The definite integral of a sum is the sum of the individual integrals In Figure 144 we setY1 25inx and Y2 X The graph of the function Y3 Y1 Y2 is shown as a bold curve We find here that the definite integral of the sum function Y3 is the sum of the definite integrals of the two functions forming the sum ntntltV1YzyXgt0y 8934802201 iquotl t 39839 Zr 1 In n quot 8934802201 IKXMFBJBHEOZZ ixdx133918022 Hxdx391 Figure 144 The de nite integral of a sum iv the sum of the integralst Constant multiples can be factored out of a definite integral We already saw an example of this with J 25inxdx 2quot sinxdx Another example is shown in Figure 145 0 0 v1axinuo V1innll Fn I nt 35 i n X y X l y 9 n 6 A A 3f nIntsinXX Oxngt V V 6 LOWtVLiMR V CY UM xo II P V0 X 29115527 V17EHE399 txdx Figure 145 A constant multiple of a function can be factored out of a de nite integral The definite integral as a function y 1 fn nt X Recall that nDeriVT2 T 1 is the derivative value at 1 and by replacing 1 with X we can create a derivative function nDeriVT2 T 1 We repeat this dummy variable technique to create an integral function We will use T as the dummy variable in the examples it could be called X and it would make no difference For example Y Seesaw ntnt cost T A X Tip Graphing functions de ned with fnlnt is quite slow setting Xres to a higher number will increase graphing speed In Figure 146 we choose 0 as the lower limit so Y1 ntnt cost T A X and we graph it using a ZTrig setting and Xres 3 We see that the graph of Y1 looks like a sine function and we check this 7139 using a table with TblStart 0 and A Tb 3 You could also graph the sine function and check that the two functions have the same graph mm mm Hotz PM THBLE SETUP X V1 V2 YiEf nIntcosT TblStart0 o o T X ATb1n12 2515 25 zsnaz VzEsiMX IndPnt r 0711 70711 V DePend Low 2560 1560 VH 1309 5559 5555 y 15705 1 1 xtnonasss 4305251 Vs X Figure 146 The integral function of the cosine appears to be the sine function If we change the lower limit in the fnlnt definition then interesting things will happen You can see from Figure 147 that the three functions are vertical shifts of one another Thinking of the derivative as the rate of change of yvalues these three should have the same derivative Each ofthese three functions is called an antiderivative of cosx The common notation used is Icosxdx sinx C where C is an arbitrary constant Neitherthe Tl82 nor 83 calculator can give you symbolic solutions of this type but the T92 does have this power rim PM PM rm Iaf ggntkosAT VEEf nInUcoMD yTy11X Va f ggntcosT 1V2 x1oassa v1oasisos Figure 14 7 Antiderivatives of the cosine mcrion are of the farm sinx C

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