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The -17 Annual Integration Games of Improper Integrals, Integration By Parts, and Partial Fractions Written by Zachary Trego and Sarah Hyman http://prawnandquartered.com/tag/contest/ 1 We hope you find this book helpful in learning these difficult forms of integration, and we hope you enjoy our integration of the Hunger Games (Pun Intended) 2 Table of Contents • General Overviews Integration by Parts .……………………………………………………………………p 4 Partial Fractions …...…………………………………………………………………… p 5 Improper Integrals ………………………………………………………………………..p 6 • Graphing calculator examples ………………………………………………………………….p 7 • Analytical Examples Integration by Parts ………………………………………………………………………p 8,9 Partial Fractions ……………………………………………………………………………p 10,11 Improper Integrals ………………………………………………………………………..p 12 • AP Conceptual Example ……………………………………………………………………………p 13,14 • A Nice Break …………………………………………………………………………………………….p 15-21 • AP Multiple Choice Example ……………………………………………………………………..p 22-24 • AP Free Response …………………………………………………………………………………….p 25-27 • Real World Applicability ……………………………………………………………………………p 28 • Contributing Mathematician ……………………………………………………………………..p 29 • Analytical Examples …………………………………………………………………………………… P 30-33 • AP Multiple Choice ……………………………………………………………………………………p 34-38 • AP Free Response ……………………………………………………………………………….......p 39 • Works Cited …………………………………………………………………………………………….. P 40 3 Integration by Parts Overview = − This process should be utilized when integrating a function that contains a cyclical 5 portion. For example, in 0 x , the cyclical portion is . This is an important skill to have, because these integrals cannot be evaluated using normal means, including u substitution. This should only be used for special cases that lend themselves to integration by parts. This process can be used to integrate other functions that may not lend themselves to integration by parts, but this is usually not necessary. Note. It may be helpful to apply Hunger Games to these problems. Let Katniss be K and Peeta be P as they are both characters with their own paths just as P and K are separate functions with their own properties. K P http://www.wallpaperpin.com/wallpaper/1680x1050/hunger-games-movie-wp-katniss-and-peeta-19576.html 4 Partial Fractions Overview Evaluate the integral : 1 2 − 1. When u substitution does not work, factor the denominator. 2. Then, separate the fraction, using different variables for each numerator. The two variables represent the constants that will make the two fractions equal to the original. 3. Finally, evaluate for each variable by substituting numerical values for x. Since x is just a variable, not a constant, we can substitute any value in for it. 4. Note that the values of P and K are equal in magnitude but opposite in sign. Though this happens from time to time, the values do not always follow such a pattern. Always do the work to determine these values!!!!!! 5 1 ( − 1) = + −1 1 = (x-1)P +(x)K Let x=0 1= -1P P= -1 Let x=1 1=1K K=1 −1 1 + −1 Improper Integrals Overview Whenever you see an integral with one or more undefined endpoints, rewrite the integral using limits before evaluating. ∞ = lim →∞ 0 0 As the function approaches infinity, the speed at which it approaches a certain value determines whether or not the integral exists or not. These integrals can either be determinate or indeterminate. An indeterminate integral would contain an undefined value or approach either -∞ or ∞. * For integrals that resemble the form , the integral converges for every value such that p > 1. This property reflects the speed at which the function approaches 0. 51 51 = lim dx 0 + →0 10 Look out for these integrals too! 0 1 = lim− →3 −3 6 0 1 + lim+ →3 −3 10 1 −3 Graphing Calculator Examples for Determinate and Indeterminate Integrals of Functions Indeterminate 1 = = Determinate 1 = 2 1 = 1 3 Notice the speed at which the functions approach the x-axis. The fuctions = = 1 3 clearly approach the x-axis much faster than the functions y= functions y= 1 1 1 1 2 and 1 = . The and y= are not integrable as x approaches infinity. Thus, if the degree of x in the denominator is greater than 1, and the numerator contains just a constant, then the improper integral is determinate.7 Analytical Example 24 Integrate the following integral 0 http://thehungergames.wikia.com/wiki/74th_Hunger_Games 8 Solution 24 1. First, using the general form of integration by parts, determine what parts of the function to designate as P and dK. Assign the part that will become a constant when differentiated as P. Assign the easily integrable part as dK. In this example, x will be P and will be dK. 2. Then, take the derivative of P, and take the integral of dK. 3. Lastly, use P, K and dP in the general form of integration by parts to evaluate the integral. 0 P=x dK= dP=1 dx K= 24 = x 0 24 1 0 ( xe e ) x x - 24 0 24e 24 e 24 (0 1) 9 Analytical Example Throughout the Hunger Games, Katniss and Peeta are forced to separate in order to survive, but they eventually reunite to achieve victory. This parallels the process by which you solve partial fractions, because you must separate the function in order to achieve integration victory. Also, you can plug in any value for x in order to solve for K and P, because the environment, which is variable, in which the Hunger Games takes place can be anything from a desert to a tundra. If Katniss and Peeta are in the Integral Games for 18.358 days, and their journeys are modeled by the 5 function f(x)=142 −6−44 . What is the total displacement they traveled during the time that they were there? 5 Obviously, the function factors into (7+11)(2−4) . 10 Solution Once the denominator is factored, separate the fraction into two pieces. Then, multiply both sides of the equality by the original denominator. Figure out which x values will have each term, P and K, go to zero, because having one term go to zero allows one to easily find the value of the other constant. Plug in the values for P and K into the original two separated fractions, and integrate each fraction to eventually solve for the original integral. 18.358 5 0 14 2 −6−44 = 7+11 + 2−4 5 = 2x − 4 P + 7x + 11 K 2 Let x=2 10= 25K K=5 Let x= 18.358 5 0 14 2 −6−44 = 11 = −50 7 11 P=10 2 18.358 11 10 5 ( + )dx 0 7+11 2−4 18.358 11 1 ln(7 x 11) 10 7 0 −11 −55 7 7 18.358 2 1 ln(2 x 4) 5 2 0 Analytical Example Evaluate the integral 4 1 0 −3 Solution First, determine whether there are any undefined values within the integral. At x=3, the integral is undefined. Then, separate the integral into two in order to evaluate the integral from 0 to 3 and from 3 to 4. You can do this, because there is no area at a single point. Thus, separating the integral accounts for the undefined value at x=3 and does not alter the area under the curve. If either one of the separate integrals is indeterminate, then the 12 entire integral is indeterminate. 3 1 0 −3 + 4 1 3 −3 ln (x-3) + ln (x-3) ln x 3 ln x 3 3 4 0 3 AP conceptual example x 4 6 8 10 12 F(x) 1 12 1 32 1 60 1 96 1 140 14 1 192 16 1 252 The function F(x) is continuous, differentiable, and constantly decreasing. ∞ Based on the values in the chart, determine whether or not 4 is determinate or indeterminate. Justify your answer. 13 Solution 6 −(4) 6−4 ***** 8 − 6 8−6 Show that > or the equivalent using other points. Since both the values and the slope of the secant lines between values are decreasing, one can determine that the values of the function are approaching 0 fast enough for the ∞ function to converge. Thus, 4 is determinate. 14 15 http://missjordennesclass.wordpress.com/2013/03/20/what-do-you-think-of-thispuppy-its-called-a-pomsky-pomeranian-and-husky/ http://imgur.com/od0u7km http://www.rarely-pins.com/tag/husky/ http://weheartit.com/entry/25629224 16 17 http://www.telegraph.co.uk/earth/earthpicturegalleries/8280986/Polar-bear-pictures.html?image=1 http://cutestuff.co/2011/08/newborn-tiger-cubs/ 18 http://coolpets4u.blogspot.com/2012/04/kittens-and-puppies-pictures.html 19 http://www.multyshades.com/2012/06/40-heartwarming-examples-of-baby-animal-photography/ 20 21 AP Level Multiple Choice Example If 3 2 2 +6−8 = Yes, you have to simplify. Why, you ask? We want to see you suffer. , f(x)= 3 (A) (2−2)10 6 (+4)5 + 3 10 6 5 (B) (2 − 2) ( + 4) (C) 6 5 (2 − 2)( + 4) + (D) 3 4 2 2 + 6 − 8 + (E) + 3 2 x 2 2 3 x 3x 2 8 x 3 22 http://thehungergames.wikia.com/wiki/Peeta_Mellark Answer and Solution 3 = 2 2 + 6 − 8 3 = + (2 − 2)( + 4) (2 − 2) ( + 4) 3 = + 4 + (2 − 2) 3(−4) = −4 + 4 + (2(−4) − 2) 3(1) = (1) + 4 + (2(1) − 2) P= 3 5 K= = = 3 10 2 − 2 + 3 3 = 2 2 + 6 − 8 5 6 5 +4 → 1 6 + 2 − 2 5 1 +4 3 10 = (2 − 2) ( + 4) Therefore, the answer to this problem is B 23 6 5 6 5 Why You Were Wrong Choice A: You subtracted instead of added. Before simplifying, you got ln (2 x 2) 3 10 ln ( x 4) 6 5 +c Be careful with sign changes. 3 Choice C: You made a mistake with the chain rule. When integrating 5 1 2−2 6 you multiplied by two instead of dividing by two. You got 5 ln (2 x 2) + c 6 This is why you thought you could factor out the 5 . Choice D: You made a number of errors. You thought you could use u substitution, so you made u=2 2 +6x+8. In this case, du= 4x+ 6, but you forgot the +6. Thus, you solved 3 4 1 . Choice E: If you got this answer, we would suggest retaking AP Calculus AB. This is just as stupid as leaving a bag of apples hanging above land mines near your supplies, like the player from District 3. 24 AP Level Free Response (Calculator) While Katniss Everdeen is traveling through the woods, a fire ball explodes next to her. Consequently, she is injured and cannot walk. The fire ball generates a forest fire that is spreading toward Katniss at a rate, in meters per minute, modeled by the function = , where t is time in minutes. Katniss is able to limp at a rate of 6 meters per minute toward a river near by. If the fire reaches the river in minutes, will Katniss reach the river in time? To practice integration by parts, only use a calculator to find the actual answers after integrating. 25 http://mockingjay.net/2012/04/06/new-hq-still-katniss-running-through-fire/ Solution dx 0 = = = = − = − − 0 = = = = − 0 = − + − 0 0 2 0 − = (-e cos x e sin x) 0 x x cos + sin + 0 cos 0 − sin(0) +1 2 = 12.070 meters 26 Solution Continued… 6 dx 0 = 6 x0 = 6 − 6 0 = 18.850 meters Katniss traveled 18.850 meters, while the fire traveled 12.070 meters in the same amount of time. Thus, Katniss outran the fire to reach the river some time t before time t= minutes. Be careful when solving this type of problem. Since it involves integration by parts, there are a lot of steps involved. If this appears on the AP test, be sure to show every calculus step that you took to get your answer. 27 Real World Applicability Improper Integrals commonly pop up when dealing with probability. Integrating a function to infinity can model the probability of an event as the probability approaches 100%. Integration by parts is an important tool in the field of engineering. Integration by parts is needed in common problems, including electric circuits, heat transfer, vibrations, structures, fluid mechanics, transport modeling, air pollution, and electromagnetics. Although partial fractions do not have a specific applicability, they are useful in calculating numerous integrals. The ability to integrate more functions allows one to be able to solve more calculus problems. All three of these integration skills have virtually endless applications in the real world. Integration is involved in, but not limited to, finding the area bounded by curves, finding the volume of solids of revolution, finding the center of mass, finding moments of inertia, calculating work done by a variable force, and finding average values. 28 Archimedes Archimedes, known as “the wise one,” “the master,” and “the great geometer,” was born in 287 B.C. in the port of Syracuse, Sicily. According to ancient Greek biographer Plutarch, Archimedes achieved so much fame because of his relation to King Hiero II and Gelon (son of King Hiero II). He was a close friend of Gelon and helped Hiero solve complex problem with extreme ease, utterly amazing his friend. His greatest accomplishments were in his utilization of integration. Archimedes was able to calculate areas under curves and volumes of certain solids by a method of approximation, called the method of exhaustion, based on using known areas and volumes of rectangles, discs, etc. His results were usually expressed, not in absolute terms, but in terms of comparisons of volumes. For instance, he could describe shapes by saying that there is a sphere of radius r surrounded exactly by a circular cylinder of radius r and height 2r. Then, Archimedes showed that the volume of the sphere is two thirds that of the cylinder. Archimedes found the sum of a geometric series in such a way as to indicate that he understood the concept of limits, which relates to improper integrals. He was also a thoroughly practical man who invented a wide variety of machines, including pulleys and the Archimidean screw pumping device. 29 Note: The problems are color-coded based on their difficulties. Yellow problems are easy. Blue problems are medium. Red problem are difficult. 30 Analytical Examples Evaluate the following integrals using integration by parts. A calculator is not required to solve these problems. 1 1) 310 2 2) ( +1) 0 3) 5) 2 4 cos 5 7xtan −1 4 3 sin 2 4) 2 1 2 3 6) −1 0 31 2 3 Integrate the following integrals using the partial fractions method of integration. A calculator is not required to solve these problems. 4 7) 0 9) 1 2 + − 6 15 8 3 − 24 2 − 72 − 40 8) Note: One zero of the function = 8 3 − 24 2 − 72 − 40 is 5 8 2 − 2 + 1 10) Note: Account for the repeating part of this function. 8 11) 1 4 + 1 90 2 − 45 32 5 2 + 3 + 7 3 2 − − 10 Integrate the following integrals using the improper integrals. A calculator is not required to solve these problems. You may have to use L’Hopital to solve these problems. Watch out for undefined values! ∞ 12) 0 6 1 2 + 4 13) −∞ 1 3 ∞ 7 − 14) 0 16) 10 ∞ 17) 5 − 0 −∞ 33 2 −8 AP Multiple Choice Exercises 1) What is the sum of the following integral? A) B) 3 10 2 − 17 − 20 −30 2 − 60 (10 2 − 17 − 20)2 5 + 4 12 66 15 165 (2 − 5) +c 12 C) (5+4)165 15 (2−5)66 D) 5 + 4 +c 12 165 15 66 (2 − 5) +c E) The function is not integrable. http://www.panempropaganda.com/news/2012/5/1/katniss-and-peetas-festival-of-victory-the-most-elaborate-vi.html 34 2) Katniss is frolicking like a kitten along the x-axis, looking for catnip at a rate, in gigameters per nanosecond, modeled by the function = , 0 ≤ t ≤ . Upon reaching the catnip, Peeta steals it, because he mistakes it for pita bread and runs in the opposite direction. How far from her starting point has she frolicked when Peeta takes the catnip and she begins frolicking backwards to get it? You may not use a calculator for this problem. 2 B) -2 C) −2 2 D) +2 2 35 http://www.loupiote.com/photos/6567820901.shtml http://www.myhungergames.com/the-hunger-games-console-games E) 3 http://www.catclaws.com/Certified-Organic-Catnip-8-oz-Bag/productinfo/1480/ A) 3) Which of the following integrals are determinate? I. 2 1 −5 8−24 II. ∞ 1 15 2 +60 III. 10 −1 A) I and II only B) III only C) I and III only D) I, II, and III E) II only http://nyulocal.com/on-campus/2012/03/23/girl-on-fire-hunger-games-review/ 36 4) Solve the following integral. 2 5 12dx A) 5 2 2 arcsin + 2 12 144 − 4 + 2 B) 5 2 2 arcsin − 2 12 4 − 144 + 2 C) 5 2 2 arcsin − 2 12 144 − 4 + 2 30 D) E) 4 + 1 − 144 http://www.fashionresister.com/2012/11/occ-metallurgy-super-nswf.html 2 −5 + 12 37 4 9 Evaluate the following integral. 1 A) 8ln4-3.75 B) -6.75 C) 72ln4+33.75 D) 72ln 4-33.75 E) 72ln4-3 http://www.dragoart.com/tuts/10222/1/1/how-to-draw-hunger-games,-the-hunger-games-logo.htm 38 AP Free Response During Katniss’s journey through the Integral Games, several other players try to kill her with their superior calculus. In order to escape her threatening competitors, Katniss decides to disturb a tracker jacker hive. Although she succeeds in repelling her attackers, she is stung multiple times. Luckily, four hours after she is stung, Rue removes the stingers and applies anti-venom. The rate at which the tracker jacker venom is entering Katniss’s body, in milliliters per hour, is modeled by the function ′ = − + , 0 ≤ t ≤ 4. The rate at which the anti-venom neutralizes the venom, in milliliters per hours, is modeled by the function ′ = − , t ≥ 4. integral − ( (a) Evaluate the − + ) . Using correct units explain the meaning of your answer. (b) At what time after Katniss gets stung has all of the venom in her body neutralized? (c) If Rue did not show up to save Katniss, at what time would Katniss have died? Note: 20 ml of venom is deadly. http://thehungergames.wikia.com/wiki/Tracker_jacker 39 Works Cited "Archimedes." Ancient Greece. N.p., 2003-2012. Web. 03 June 2013. "Archimedes of Syracuse." JOC/EFR, Jan. 1999. Web. 03 June 2013. Bourne, Murray. "Applications of Integration." Interactive Mathematics. N.p., 24 Aug. 2012. Web. 03 June 2013. Burt, Brandon. "Re: What Are the Uses of Improper Integrals/calculus?" Web log comment. Answers.yahoo.com. Yahoo Answers, 2008. Web. 3 June 2013. Divo, Eduardo. Integration by Parts Applications in Engineering. Rep. N.p.: n.p., 2009. Print. Khamsi, Mohamed A. "Convergence and Divergence of Improper Integrals." Convergence and Divergence of Improper Integrals. SOS Mathematics, 3 Dec. 1996. Web. 03 June 2013. Petrov Petrov, Yordan. "The Origins of the Differential and Integral Calculus 1." Math10.com. N.p., 27 Sept. 2005. Web. 03 June 2013. Simmons, G. "History of Calculus." N.p., 1985. Web. 03 June 2013. 40