x x 1 1. F x 1 t dt 2. F x dt 3. F x 1 t dt t x x 2x x F x dt 8. F x dt 9. F x sin x 1/ x cos x 10 x

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1 Pre-Calculus - Honors Secion 8. Uni 8 More on Inegraion Find F' Fundamenal Theorem of Calculus No Calculaor. F d. F d. F d. F d 5. F cos d. F d 7. F d 8. F d 9. F d sin / cos. F d. F sin d. F d. F d. F d 5. F d Evaluae each of he following definie inegrals.. 5 d 7. d 8. d / cos 9. sin d. cos d. d / sin 7 5. d. d. d 5. d. d 7. d d 9. 9 d. 8 d. d. d. d Find he area of he region bounded by he -ais, he given curve y = f(), and he given verical line(s).. y,, 5. y,, 8. y, 7. y 8. y,, 9. y,,. y,,. y,,. y,,. Find he area bounded by he coordinae aes and he line + y =. Find he area beween he curve y, he -ais, and he lines = and =. 5. Find he area beween he curve y and he y-ais. (Hin: Inegrae wih respec o y.) Pre-Calculus - Honors Secion 8. Uni 8 More on Inegraion Subsiuion in Definie Inegrals No Calculaor Evaluae he following definie inegrals. /. d. d. an sec d

2 . d 5. d. d 9 7 / / 7. d 8. cos sin d 9. cos sin d cos sin. d. d. d sin / cos / cos 5. cos d. d 5. 5 d /9 / / /. d 7. cos sin d 8. an sec d Find he area bounded by he -ais, he given curve y = f(), and he given verical lines. 9. y,,. y,,. y,,. y,,. y,,. y,, 5. y,, 7. y,, 7. y 9,, 8. y,, 9. y 9,,. y,,. Find he area beween he curve y and he coordinae aes in he firs quadran.. Find he area conained beween he -ais and one arch of he curve y cos. Pre-Calculus - Honors Secion 8. Uni 8 More on Inegraion Area Beween Curves No Calculaor Se up, bu do no evaluae, he definie inegral(s) necessary o find he area of he regions enclosed by he given curves and lines.. The curve y and he line y =.. The -ais and he curve y. The y-ais and he curve y y. The curve y and he line = 5. The curve y and he line y = -. The curve y and he line y = 7. The curve y y and he line + y = 8. The curve y and y 9. The curve y and he line = y +. The curve y and he line y = 8. The curves y and y. The curve y and he line y =. The curve y and he line y =. The curve y and he line =. 5. The curves y and y. The curves y and y 7. The line y = and he curve y

3 8. The curves y 7 and y 9. The curves y and y. The curves y and y. The curves y = cos and he line y = -, on,. The curves y sec, y an, and he lines and. Find he area of he riangular region in he firs quadran bounded by he y-ais and he curves y sin and y cos.. Find he area of he region in he firs quadran bounded by y,, and y 5. Find he area of he region bounded on he righ by + y =, on he lef by y, and below by he -ais. Pre-Calculus - Honors Secion 8. Uni 8 More on Inegraion Ne Disance vs. Toal Disance No Calculaor For each scenario below: a. Find he ne disance raveled on he given inerval. b. Se up, wihou absolue value, he inegral or inegral epression required o compue he oal disance raveled.. v 5cos.. s v a cos, v s v sin 7. s sin 9. a cos, v (omi b) 8. a 9.8 sin,v.9 (omi b) (omi b). a, v

4 Secion Approimaing Definie Inegrals The definie inegrals you can compleed husfar would be considered convenien. They all involve inegrals which you can deem as possible. In many cases, however, funcions do no have aniderivaives. Noneheless, we are asked o find he area under such curves. A mean by which we can esimae such areas is called Riemann sums. There are hree such Riemann sums for which you are responsible: Lef hand sum adds he areas of recangles whose heigh is he lef side of he recangle. Righ hand sum - adds he areas of recangles whose heigh is he righ side of he recangle Midpoin sum adds he areas of recangles whose heigh is he heigh of he midpoin of an inerval Trapezoidal Rule compues he area of rapezoids and adds hem ogeher. Take for eample, he inegral 8 d canno be evaluaed by any mehods which we know. However, we can creae an approimaion of he area under he curve / from o 8. For his eample, we will use four subinervals. We begin wih a graph of he funcion and a able of values shown o he righ of he graph. Boh he graph and able are obained from he graphing calculaor. / From his informaion, we can compue all four approimaions of he inegral (area under he curve). For he purposes of his eample, he widh of each figure is one, bu we will wrie in he in he even ha he widh of he figure is NOT. Lef Hand Sum We compue and add he areas or recangles whose heigh is he lef segmen of each recangle. These recangles are shown below. Righ Hand Sum We compue and add he areas or recangles whose heigh is he righ segmen of each recangle. These recangles are shown below. 8 8 A A Noe ha his is an OVEResimaion in his case. Noe ha his is an UNDEResimaion in his case. Midpoin Sum We compue and add he areas or Trapezoidal Rule We compue and add he areas of all recangles whose heigh is he midpoin of each inerval. Four rapezoids. Remember he area of he rapezoid is These recangles are shown below. ½ he heigh imes he sum of he bases so he middle hree bases will be couned wice. Noe: A graph will be provided on he board, bu is oo difficul o accuraely recreae here. The compuaion is: A A

5 Pre-Calculus - Honors Secion 8.5 Uni 8 More on Inegraion Riemann Sums Esimaing he Definie Inegral Calculaor Required. Approimae d using lef hand sums and n = 5.. Approimae. Approimae. Approimae d using midpoin sums and n =. d using he rapezoidal rule and n =. d using a righ hand sum wih n = Approimae d using a righ hand sum wih n =. 5. Approimae d using he rapezoidal rule wih n =. 7. Approimae 7 8. Approimae d using a midpoin sum wih n =. d using a lef hand sum wih n = The able below gives velociy a various ime inervals. Esimae he oal disance raveled from = o = using: a. he rapezoidal rule wih n = b. he lef hand sum wih n = c. he righ hand sum wih n = d. midpoin sum wih n = Time () Velociy (v) Use he midpoin sum wih n = o esimae he value of. Use he able of values below o esimae a. he lef hand sum wih n = b. he midpoin sum wih n = c. he rapezoidal rule wih n = 5. Use he able of values below o esimae f d using: f d. 8 f() 8 5 f() v d using he rapezoidal rule. Time () Velociy (v) 8 9

6 Pre-Calculus - Honors Uni 8 More on Inegraion Uni 8 Review *** - Calculaor Required Muliple Choice Review YEAH!!!! ***. If d is approimaed by hree inscribed recangles of equal widh, hen he approimaion is: A. B. C. 8 D. 8 E. 7 for f, hen f d is a number beween elsewhere A. and 5 B. 5 and C. and 5 D. 5 and E. and 5. If. d is: A. B. C. D. E. none of hese. If f d, hen f d A. B. C. 9 D. E. 5 for f. Then sin for A. ½ B. C. k d, hen k = 5. Le f be he funcion defined by. If 7. f d D. A. B. C. D. E. d A. B. C. ½ D. E. b b d, hen b = 8. If A. B. C. D. E Suppose F d for all real, hen F (-) = A. B. C. / D. E. / F sin d. The value of F F' is. Consider he funcion F defined so ha A. B. C.. If f and g are coninuous funcions such ha g' f for all, hen E. D. E. f d A. g' g' B. g' g' C. g g D. f f E. f ' f '. If F cos d, hen F' A. sin + B. cos + C. sin D. sin E. sin +

7 f if and le F f d. if Which of he following saemens are rue? I. F() =.5 II. F () = III. F () = A. I only B. II only C. I and II only D. II and III only E. I, II III. Le. Le funcion f() be always increasing and always concave up on a closed inerval [a, b]. The inerval [a, b] is pariioned ino n equal lengh subinervals and hese are used o compue a righ sum (U), a lef sum (L), and he rapezoidal rule approimaion (T). If I f d, which saemen below is rue? b a A. L< U < T < I B. L < U < I < T C. L < T < I < U D. L < I < T < U E. L < I < U < T Free Response Show he work ha leads o your answer dy 5. Solve he differenial equaion d. Find an equaion for he curve whose slope a he poin (, y) is poin (, -). 7. Evaluae: 9. Evaluae: subjec o he iniial condiions y = when =. cos sin d 8. Find df if F d d, if he curve is required o pass hrough he d. Solve he differenial equaion dr r ds s. A paricle moves wih acceleraion a. Assuming ha he velociy v = / and he posiion s = -/5, boh when =, find he velociy (v) and posiion (s) funcions.. The acceleraion of graviy near he earh s surface is f / sec. A sone is hrown upward from he ground wih a speed of 9 fee per second. How high will he sone rise in seconds? How high will he sone go?. Evaluae: 5. Evaluae u / df du. Find if F u d d. Evaluae: sin d sec d ***7. Some of he values of a coninuous funcion are given in he able below. Find he approimaion of f d using each of he four mehods (lef-hand sum, righ-hand sum, rapezoidal rule, and midpoin sum) using five inervals f()