4.3 The Fundamental Theorem of Calculus

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1 Clculus Mimus 4. The Fundmentl Theorem of Clculus We ve lerned two different rnches of clculus so fr: differentition nd integrtion. Finding slopes of tngent lines nd finding res under curves seem unrelted, ut in fct, they re very closely relted. It ws Isc Newton s techer t Cmridge University, mn nme Isc Brrow (60 677), who discovered tht these two processes re ctully inverse opertions of ech other in much the sme wy division nd multipliction re. It ws Newton nd Leiniz who eploited this ide nd developed the clculus into it current form. The Theorem Brrow discovered tht sttes this inverse reltion etween differentition nd integrtion is clled The Fundmentl Theorem of Clculus. The lunr crter Brrow is nmed fter Isc Brrow. The Wheel Brrow is not. We re now redy for the shortcut rule for integrtion. This is wht we ve een witing for: n esier wy to clculte definite integrls. The Fundmentl Theorem of Clculus, Prt (FTOC) If f is continuous on, nd F is n ntiderivtive of f, then This integrl gives us the NET chnge!!! f d F F F Emple : Evlute the integrls using the FTOC. () e d () / 0 d (c) e d Emple : Evlute ech definite integrl using ny method. Verify on the clcultor. () d () 4 d (c) /4 0 sec d (d) 0 d Pge of 9

2 Clculus Mimus Emple : / e 4 Find the ect vlue of 0 / / 5 d Just s we cn use res of regions to help us find definite integrl vlues, now tht we (officilly) know how to evlute definite integrls, we cn use integrls to help us find res. But we must use cution, for re is lwys POSITIVE, nd WE re responsile for mking negtive regions positive!! In such cses, we must CLEARLY IDENTIFY THE INDICATED REGION! Emple 4: Find the re ounded y the prol () Without clcultor y nd y 0 from 0 to () With clcultor Emple 5: Find the re of the region ounded y the curves y 0, y, nd e () Without clcultor () With clcultor Pge of 9

3 Clculus Mimus Emple 6: Without clcultor, find the re of the region ounded y the -is nd the function y intervl. Use the symmetry of the function to help you evlute the integrl. 0 on the The second prt of the theorem dels with integrl equtions of the form F f t dt where f is continuous function on,, nd vries etween nd. Notice tht this integrl eqution is function of, which ppers s the upper limit of integrtion. If let,, then we cn define f t F s the re under the curve from to. hppens to e positive, nd we Emple 7: The grph of f t is given. Let F f t dt. Use the res of the regions to find the following: () () F F 9 (c) F 5 (d) F 8 (e) F 0 Pge of 9

4 Clculus Mimus Emple 8:, find simplified, epnded version of F If () F t t dt Once you find F, find its derivtive, F. Wht do you notice? y evluting the definite integrl. The Fundmentl Theorem of Clculus Prt (FTOC) specil cse If f is continuous function on,, then the function F defined y F is continuous on, nd differentile on f t dt,,. Additionlly, F f d d f tdt f. We cn lso sy tht Emple 9: Evlute () d d sect dt d () d 5 p sin p dp (c) k d e k k d ln k dk Chin Rule nyone? Pge 4 of 9

5 Clculus Mimus The FTOC, most generl form: g d f tdt f g g f h h d h Emple 0: Evlute the following using the FTOC, then verify y doing in the Loooooooong wy. 7 d () sec t dt d d () t 5t dt d e Emple : If ln t dt, find F F sin The Men Vlue Theorem (for Integrls) If f is continuous on the closed intervl,, then there eists numer c, such tht f d f c in the CLOSED intervl Where f c is clled the verge vlue of the function f on the intervl, f c. cn e eplicitly solved for. The ove eqution ove Pge 5 of 9

6 Clculus Mimus f c f d or f c f d Emple : F (Clcultor) In New Brunfels, the temperture (in ) t hours fter 9.m. ws modeled y the function T t. () Find the verge temperture during the -hour period from 9.m. to 9 p.m. 50 4sin t Show your integrl set up! () find the vlue(s) of t t which the MVT gurntees the temperture in N.B. ws the verge temperture. Emple : Find the vlue of c gurnteed y the MVT for integrls for f on, grphiclly.. Interpret the result Emple 4: Show tht the verge rte of chnge of cr s position over time intervl t t verge vlue of its velocity function over the sme intervl., is the sme s the Pge 6 of 9

7 Clculus Mimus Emple 5: The tle elow gives vlues for continuous function. Using the vlues given, find the rithmetic men of f. Using trpezoidl pproimtion using 6 suintervls, estimte the verge vlue of f on 0,50. For this continuous function, is it possile to determine which is more ccurte? Is one method etter prctice thn the other? f Sometimes we might hve to solve n integrl eqution! Being le to simplify definite integrls with vriles in the intervl of integrtion is importnt. Here re couple of emples showing n importnt ppliction tht is importnt. Emple 6: Find the numer(s) such tht the verge vlue of f 6 on the intervl 0, is equl to. Emple 7: Solve the following conditionl equtions for the indicted vrile. () t 7tdt 0, for () 5 k 4 d 0, for k Pge 7 of 9

8 Clculus Mimus IMPORTANT Emple 8: f 0 5 If nd f, find f y () finding the prticulr solution to the differentil eqution, then evluting the solution t, nd then y () using definite integrl. In the previous emple, the first method relied hevily upon our ility to find the ntiderivtive of the integrnd. This is not lwys esy, possile, or prudent! Being le to epress prticulr vlue of prticulr solution to derivtive s definite integrl is of prmount importnce, especilly when we don t know how to find generl ntiderivtive. Hrd Fcts To Refute: A. Where you re t ny given time is function of ) where you strted nd ) where you ve gone from your strting point (displcement). B. Wht you hve t ny given moment is function of ) wht you strted with plus ) wht you ve ccumulted since then. When you ccumulte t vrile rte, you cn to use the definite integrl to find your net ccumultion. Importnt Ide of Accumultion********************************(* mens VERY IMPORTANT) Wht I hve now = Wht I strted with + Wht I ve ccumulted since I strted This cn e epressed mthemticlly s f f f d Pge 8 of 9

9 Clculus Mimus Emple 9: If f 4sin nd f, using your clcultor, () through (c) only, find the following. Be sure to show your INTEGRAL SET UP for prts () through (c). () 5 f () f (c) f (d) n integrl eqution for f Emple 0: (Clcultor permitted) Rw sewge is leking from storge tnk t rte given y the eqution S t t e, where S is mesured in gllons per hour, nd t is mesured in hours. If t t were 57 gllons of rw sewge in the tnk, () How much sewge hs leked out of the tnk from t hours to t 5 hours? hours there () How much sewge ws initilly in the tnk when it strted leking t t 0 hours? (c) Write n integrl eqution tht gives the mount of rw sewge in the tnk, At, t time t. Emple : (Clcultor Permitted) At the strt of Christms Brek, t t 0 dys, mn weighed 80 pounds. If the mn gined weight during the rek (I m not mentioning ny nmes... or even sying tht it s me... OK, t I hve wekness for fruit cke...) t rte modeled y the function W () t 0sin pounds per dy, 8 () Wht ws the mn s weight (in pounds) t the end of the rek, 4 dys lter? Show your integrl set up, nd don t judge. () At wht time ws the mn t his heviest during the rek? Wht ws his weight t this time? Impressive? Show the food work tht leds to your nswer. Pge 9 of 9