Area Approximatio ad Accumulatio Studet should be able to: Recogize that a defiite itegral gives a accumulatio or total Always give meaig to the itegral i CONTEXT to the problem Give the uits of measuremet Referece the limits of itegratio with appropriate uits i the cotext of the problem edig time Solve applicatio problems ivolvig Amout Rate dt or begiig time time2 time2 " " " " time time Curret Amout Iitial Amout rate i dt rate out dt Traslate a defiite itegral ito the limit of a related Riema sum Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org
Multiple Choice. x Let f be the fuctio give by. is the value of the left Riema sum approximatio for (A) (B) 6 (C) 62 (D) 2 f x If four subitervals of equal legth are used, what f ( x) dx? 2. (calculator ot allowed) The graph of the fuctio f is show above for x. Of the followig, which has the least value? (A) f ( x) dx (B) Left Riema sum approximatio of (C) Right Riema sum approximatio of (D) Midpoit Riema sum approximatio of f ( x) dx with subitervals of equal legth f ( x) dx with subitervals of equal legth f ( x) dx with subitervals of equal legth (E) Trapezoidal sum approximatio of f ( x) dx with subitervals of equal legth Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org 2
. (calculator ot allowed) If three equal subdivisios of, 2 are used, what is the trapezoidal approximatio of x 2 e dx? 2 (A) (B) (C) (D) (E) 2 2 e e e 2 e e e 2 2 e 2e 2e e ( 2 2 ) 2 e e e e ( 2 2 2 2 ) 2 e e e e. (calculator allowed) The fuctio f is cotiuous o the closed iterval 2,8 ad has values that are give i the table above. Usig the subitervals 2,5, 5,7, ad 7,8, what is the trapezoidal approximatio of f x dx? 2 (A) (B) (C) 6 (D) 9 (E) 2 8 5. k lim k k lim k 2 2k lim k 2k 2 lim k 5 x dx Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org
6. (calculator allowed) The graph of f, the derivative of the fuctio f, is show above. If the followig must be true? f I. f II. f 2 f III. f f f, which of (A) I oly (B) II oly (C) III oly (D) I ad II oly (E) II ad III oly 7. (calculator allowed) f x dx f x dx y f x Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org
t a( t) a( t) t t 9. x.5.5 2 f x 6 2 28 The table above gives selected values for a cotiuous fuctio f. If f is icreasig over the closed iterval,2, which of the followig could be the value of 2 f ( x) dx? Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org 5
. t (miutes) 2 5 9 R(t) (gallos / miute).6 5..7. A tak cotais 5 gallos of water at time t 2 miutes. Water is flowig ito the tak at a rate R( t ), where R( t ) is measured i gallos per miute, ad t is measured i miutes. Selected values of R( t) are give i the table above. Usig a right Riema sum with three subitervals ad data from the table, what is the approximatio of the umber of gallos of water that are i the tak at time t miutes? (A).6 (B) 2.9 (C) 75.6 (D) 77.9 Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org 6
Free Respose. (calculator ot allowed) t (miutes) r( t) (feet per miute) 2 5 7 2 5.7. 2..2.6.5 The volume of a spherical hot air balloo expads as the air iside the balloo is heated. The radius of the balloo, i feet, is modeled by a twice-differetiable fuctio r of time t, where t is measured i miutes. For t 2, the graph of r is cocave dow. The table above gives selected values of the rate of chage, r ( t), of the radius of the balloo over the time iterval t 2. The radius of the balloo is feet whe t 5. (Note: The volume of a sphere of radius r is give by V r.) (c) Use a right Riema sum with the five subitervals idicated by the data i the table to 2 2 approximate r( t) dt. Usig correct uits, explai the meaig of r ( t ) dt i terms of the radius of the balloo. (d) Is your approximatio i part (c) greater tha or less tha your aswer. 2 r( t) dt? Give a reaso for Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org 7
2. B t B( t) v( t) t t(secods) B( t) (meters) m v( t) sec 6 6 v( t) dt v( t) dt Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org 8
t (miutes) 2 5 8 2 v ( ) A t (meters / miute) 2 5 Trai A rus back ad forth o a east-west sectio of railroad track. Trai A s velocity, measured i meters per miute, is give by a differetiable fuctio v A (t), where time t is measured i miutes. Selected values for v A (t) are give i the table above. (c) At time trai A s positio is meters east of the Origi Statio, ad the trai is movig to the east. Write a expressio ivolvig a itegral that gives the positio of trai A, i meters from the Origi Statio, at time Use a trapezoidal sum with three subitervals idicated by the table to approximate the positio of the trai at time Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org 9
t 2 5 6 (miutes) C( t) (ouces) 5. 8.8.2 2.8.8.5 Hot water is drippig through a coffeemaker, fillig a large cup with coffee. The amout of coffee i the cup at time t, t 6, is give by a differetiable fuctio C, where t is measured i miutes. Selected values of C( t ), measured i ouces, are give i the table above. (c) Use a midpoit sum with three subitervals of equal legth idicated by the data i the table to 6 6 approximate the value of C( t) dt 6. Usig correct uits, explai the meaig of ( ) 6 C t dt i the cotext of the problem. Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org
Area Approximatio ad Accumulatio Referece Page Left ad right Riema sums Correct justificatio for over ad uder approximatios: f(x) Left Riema Sum Right Riema Sum Icreasig (f '(x) > ) Uder approximates the area Over approximates the area because f(x) is icreasig because f(x) is icreasig Decreasig (f '(x) < ) Over approximates the area Uder approximates the area because f(x) is decreasig because f(x) is decreasig Icorrect Reasoig: The left Riema Sum is a uder approximatio because the rectagles are all udereath or below the graph. Statig that the rectagles are below the fuctio is ot acceptable mathematical reasoig. It merely restates that it is a uder approximatio but does ot explai WHY. Trapezoidal approximatios Over/Uder Approximatios with Trapezoidal Approximatios f(x) Trapezoidal Sum Cocave Up (f ''(x) > ) Over approximates the area because f ''(x) > Cocave Dow (f ''(x) < ) Uder approximates the area because f ''(x) < Copyright 26 Natioal Math + Sciece Iitiative, Dallas, Texas. All rights reserved. Visit us olie at www.ms.org