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1 Improper Integral ff() dddd aa bb, ff() dddd, ff() dddd e, d An improper integral is a definite integral that has. an infinite interval of integration.. They have a discontinuity on the interior of the interval of integration 3. Both ) and ) They are evaluated by rewriting the integral as a proper integral and then using its. Not every integral equals a finite number. In fact, you d probably epect anything integrated to or from infinity will be infinite. An improper integral that equals a finite value is said to converge to that value. An improper integral that does not equal a finite number is said to diverge. bb iiii ff() iiii cccccccccccccccccccc oooo [aa, ) ttheeee ff() dddd = ff()dddd pppppppppppppppp tthiiii llllllllll eeeeeeeeeeee bb aa aa bb bb iiii ff() iiii cccccccccccccccccccc oooo (, bb] ttheeee ff() dddd = aa aa ff()dddd pppppppppppppppp tthiiii llllllllll eeeeeeeeeeee No it we say: improper integral diverges Fact: pp ssssssssssss iiiiiiiiiiiiiiiiii aa iiii aa >, ttheeee pp dddd iiii cccccccccccccccccccc iiii pp > aaaaaa dddddddddddddddddd iiii pp. iiii aa = aaaaaa pp >, ttheeee pp dddd cccccccccccccccccc tttt pp Let ff() = for <, and let R be the unbounded region in the first quadrant below the graph of f. Find the volume of the solid generated when R is revolved around the -ais. (Note: The region is known as Gabriel s Horn or Torricelli s Trumpet.)

2 Eamples. Evaluate d. d converges to d = = d = + =. Evaluate d. d diverges d = d = { } = { } = 3. Evaluate e d. e d converges to e d = e d = { e } = { e e } = =. Evaluate ln d. ln d diverges ln d = ln d = { ln } = [{ } ln { ln } ] = { (ln ) } + = 5. Evaluate d (ln ) + = + = e. e d converges to e d = = e d { e } = e e e =

3 3 e e e e. By L Hospital s Rule, e = = e. Thus, e e = ( ) ( ) =. e e Sometimes, an integral can be doubly improper. iiii ff() iiii cccccccccccccccccccc oooo (, ), ttheeee ff() dddd = cc aa aa ff()dddd bb + ff()dddd bb where c is any real number. Symmetry can also be used to circumvent the doubleness of the impropriety. Note as well that this requires BOTH of the integrals to be convergent in order for this integral to also be convergent. If either of the two integrals is divergent then so is this integral. cc Evaluate + d. d + converges to π + d = d + d = d (by symmetry) = + d = { arctan } = = { arctan }= arctan π arctan = = π. This might be helpful: Convergent + Convergent = Convergent Divergent + Divergent = Divergent Divergent + Convergent = Divergent Divergent Divergent = Indeterminate

4 Improper Integral with Infinite Discontinuity Integral of a function that becomes infinite at a point within the interval of integration. iiii ff() iiii cccccccccccccccccccc oooo (aa, bb] ttheeee ff() dddd = aa bb cc aa + cc iiii ff() iiii cccccccccccccccccccc oooo (aa, bb] ttheeee ff() dddd = aa bb bb cc bb aa ff()dddd cc ff()dddd iiii ff() iiii cccccccccccccccccccc oooo [aa, cc) UU (cc, bb] ttheeee bb aa cc aa bb cc ff() dddd = ff() dddd + ff() dddd When an integral is improper has a finite interval of integration, it is improper because its interval spans an infinite discontinuity (vertical asymptote). These are harder to spot, so be vigilant!! Eamples. Evaluate d. d diverges d = + d = + = + + = +. Evaluate d. d converges to d = + d = + { } = + { }= = π 3. Evaluate d. d converges to.

5 5 d = d = { arcsin } = π { arcsin arcsin } = arcsin =. Evaluate ln d. ln d converges to. ln d = + ln d = + { ln } = ( ln ) + ( ln ) = + ( ln ) + = + + ln. By L Hospital s Rule, + ln = + ln = + = + ( ) =. 5. Evaluate d. d ln ln diverges ln d = + ln d = + { ln(ln ) } = + ln(ln ) + ln(ln ) = ln(ln ) ln(ln()) = ln(ln ) ln() = ln(ln ) ( ) e e 6. Evaluate d. d converges to e e d = + e d = + { e } = + { e e }= e e = e e d converges to e.

6 6 Practice Sheet for Improper Integrals () d = () e d = (3) d = () e d =

7 7 (5) e ( ln ) d = (6) 3 9 d = (7) 6 + d = (8) d = (9) e d =

8 8 () + d = () + d = () e ln d =

9 9 arctan (3) d = + e () d = ln e (5) d = (ln )

10 Arc Length If we walk along a curved path with a pedometer or a GPS device, we have a pretty good idea of how far we ve gone. If we walk along a curved path and have only the equation of the function along whose path we travel, a much more likely scenario, then we can use calculus to find how far we ve gone....if we wanted to. Oh, we want to. We can approimate our distance by dividing our path into several equal partitions and sum the distance between consecutive points. You already know where this is going... to achieve better and better approimations, we take smaller and smaller line segments. Voilà! The it process emerges once again. This finite process, with the it attached becomes a very simple integral. Here s how it s derived live!! The arc length of functions in Cartesian plane: S = dy = The Formula: L f ' b a d 3 Calculator problem: Compute the arc length of the graph of f over [,]. L.

11 Calculus aimus WS 8.: Arc Length Name Date Period Worksheet 8. Arc Length Show all work. No calculator unless stated. ultiple Choice. ( 88 BC) The length of the curve (A) 3 y = from = to = is given by 6 + d (B) + 3 d (C) (D) π + 9 d (E) π + 9 d + 9 d. ( 3 BC) The length of a curve from = to = is given by point ( ), 6, which of the following could be an equation for this curve? (A) y = 3+ 3 (B) y = 5 + (C) y = (D) y = 6 (E) y = d. If the curve contains the 3 Page of 5

12 Calculus aimus WS 8.: Arc Length 3. (Calculator Permitted) Which of the following gives the best approimation of the length of the arc of π y = cos( ) from = to =? (A).785 (B).955 (C). (D).38 (E).977. Which of the following gives the length of the graph of = y from y = to y =? y dy (B) 6 (A) ( + ) 6 + y dy (C) + 9y dy (D) 3 + d (E) + d Page of 5

13 Calculus aimus WS 8.: Arc Length 5. Find the length of the curve described by (A) 6 3 (B) 5 3 3/ y = from = to = 8. 3 (C) 5 (D) (E) Which of the following epressions should be used to find the length of the curve y = from = to =? (A) 9 + ydy (B) 9 + ydy (C) y dy (D) + y dy (E) /3 9/ + y dy Page 3 of 5

14 Calculus aimus WS 8.: Arc Length 7. (AP BC B-3) (Calculator Permitted) Let R be the region in the first quadrant bounded by the y- 3 3 ais and the graphs of y = + and y =. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the -ais. (c) Write an epression involving one or more integrals that gives the perimeter of R. Do not evaluate. Page of 5

15 Calculus aimus WS 8.: Arc Length 8. (AP BC B-) The graph of the differentiable function y f ( ) = with domain is shown in the figure at right. The area of the region enclosed between the graph of f and the -ais for 5 is, and the area of the region enclosed between the graph of f and the -ais for 5 is 7. The arc length for the portion of the graph of f between = and = 5 is, and the arc length for the portion of the graph of f between = 5 and = is 8. The function f has eactly two critical points that are located at = 3 and = 8. (a) Find the average value of f on the interval 5. (b) Evaluate ( 3 f ( ) + )d. Show the computations that lead to your answer. (c) Let g ( ) = f t 5 Eplain your reasoning. ( )dt. On what intervals, if any, is the graph of g both concave up and decreasing? (d) The function h is defined by h( ) = f length of the graph of y = h! $ # &. The derivative of h is h! " % ( ) from = to =. ( ) = f! " % $ '. Find the arc # & Page 5 of 5

16 Integral as Net Change Recall that the definite integral gives us the Net Accumulation over an interval. For things that change, we can use the definite integral to model a myriad of real-world applications.

17 Although accumulating velocities and distances is a very important application of the integral, we can accumulate oh so many other things. Here s the basic premise: If you have a rate equation that describes how something changes, the integral of that rate equation over an interval gives you the net accumulation of that something. This brings us back to this: What you have at any given moment is a combination of what you started with plus what you ve accumulated since then.

18 Sometimes you are gaining while your are losing. Think of pouring water into a bucket that has a small hole at the bottom. In this case... What you have at any given moment is a combination of what you started with plus what you ve accumulated since then minus how much you have lost since then.

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20 Sometime we variable rates of accumulation that vary within and between time intervals piecewise anyone?

21 Sometimes we just accumulate y-values and no units are involved.

22 Calculus aimus WS 8.: Integral as Net Change Name Date Period Worksheet 8. Integral as Net Change Show all work. Calculator Permitted, but show all integral set ups. ultiple Choice. The graph at right shows the rate at which water is pumped from a storage tank. Approimate the total gallons of water pumped from the tank in hours. (A) 6 (B) (C) 36 (D) (E) 8. The data for the acceleration ( ) a t of a car from to 5 seconds are given in the table below. If the velocity at t = is 5 ft/sec, which of the following gives the approimate velocity at t = 5 using a Trapezoidal sum? (A) 7 ft/sec (B) 5 ft/sec (C) ft/sec (D) 5 ft/sec (E) ft/sec Page of 9

23 Calculus aimus WS 8.: Integral as Net Change 3. The rate at which customers arrive at a counter to be served is modeled by the function F defined by t F( t) = + 6cos π for t [,6], where F( t ) is measured in customers per minute and t is measured in minutes. To the nearest whole number, how many customers arrive at the counter over the 6-minute period? (A) 7 (B) 75 (C) 73 (D) 7 (E) 756.5t e. Pollution is being removed from a lake at a rate modeled by the function y = tons/yr, where t is the number of years since 995. Estimate the amount of pollution removed from the lake between 995 and 5. Round your answer to the nearest ton. (A) (B) 7 (C) 56 (D) 6 (E) 7 rt = e million barrels per year, where t is time measured in years, for t. Which of the following epressions gives the amount of oil consumed by the country during the time interval t? 5. A developing country consumes oil at a rate given by ( ). (A) r ( ) (B) r( ) r( ) (C) rʹ ( t) dt (D) r ( t) dt (E) r ( ) t Page of 9

24 Calculus aimus WS 8.: Integral as Net Change Free Response. Show all integral set ups and include units when appropriate. 6. The temperature outside a house during a -hour period is given by πt F( t) = 8 cos, t F t is measured in degrees Fahrenheit and t is measured in hours. Where ( ) (a) Find the average temperature, to the nearest degree Fahrenheit, between t = 6 and t =. (b) An air conditioner cooled the house whenever the outside temperature was at or above 78 degrees Fahrenheit. For what values of t was the air conditioner cooling the house? (c) The cost of cooling the house accumulates at the rate of $.5 per hour for each degree the outside temperature eceeds 78 degrees Fahrenheit. What was the total cost, to the nearest cent, to cool the house for this -hour period? Page 3 of 9

25 Calculus aimus WS 8.: Integral as Net Change 7. The rate at which people enter an amusement park on a given day is modeled by the function E defined by 56 E( t) =. t t+ 6 The rate at which people leave the same amusement park on the same day is modeled by the function L defined by 989 Lt ( ) =. t 38t+ 37 Lt are measured in people per hour, and time t is measured in hours after midnight. Both E( t ) and ( ) These functions are valid for [ 9, 3] t, which are the hours that the park is open. At time t = 9, there are no people in the park. (a) How many people have entered the park by 5: P.. ( t = 7 )? Round your answer to the nearest whole number. (b) The price of admission to the park is $5 until 5: P... After 5: P.., the price of admission to the park is $. How many dollars are collected from admissions to the park on the given day? t ( ) (c) Let H ( t) = E ( ) L( ) d for t [ 9, 3]. The value of ( 7) 9 H to the nearest whole number is 375. Find the value of Hʹ ( 7) and eplain the meaning of H ( 7) and ( 7) the park. Hʹ in the contet of (d) At what time t, for t [ 9, 3] maimum?, does the model predict that the number of people in the park is a Page of 9

26 Calculus aimus WS 8.: Integral as Net Change 8. AP - Two runners, A and B, run on a straight racetrack for t seconds. The graph above, which consists of two line segments, shows the velocity, in meters per second, of Runner A. The velocity, in meters per t second, of Runner B is given by the function v defined by vt ( ) = t + 3. (a) Find the velocity of Runner A and the velocity of Runner B at time t = seconds. Indicate units of measure. (b) Find the acceleration of Runner A and the acceleration of Runner B at time t = seconds. Indicate units of measure. (c) Find the total distance run by Runner A and the total distance run by Runner B over the time interval t seconds. Indicate units of measure. Page 5 of 9

27 Calculus aimus WS 8.: Integral as Net Change 9. AP B- A particle moves along the -ais so that its velocity v at any time t, for t 6, is given by sint vt ( ) = e. At time t = (a) On the aes provided, sketch the graph of ( ), the particle is at the origin. vt for t 6. (b) During what intervals of time is the particle moving to the left? Give a reason for your answer. (c) Find the total distance traveled by the particle from t = to t =. (d) Is there any time t, < t 6, at which the particle returns to the origin? Justify your answer. Page 6 of 9

28 Calculus aimus WS 8.: Integral as Net Change. AP 6- t Lt = 6 tsin cars per hour 3 y = L t is shown above. At an intersection in Thomasville, Oregon, cars turn left at the rate of ( ) over the time interval t 8 hours. The graph of ( ) (a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval t 8 hours. (b) Traffic engineers will consider turn restrictions when Lt ( ) 5 cars per hour. Find all values of t for which Lt ( ) 5 and compute the average value of L over this time interval. Indicate units of measure. (c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than,. In every two-hour time interval, 5 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Eplain the reasoning that leads to your conclusion. Page 7 of 9

29 Calculus aimus WS 8.: Integral as Net Change. AP 8- Concert tickets went on sale at noon ( t = ) and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable function L for t 9. Lt at various times t are shown in the table above. Values of ( ) (a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:3 P.. ( t = 5.5 ). Show the computations that lead to your answer. Indicate units of measure. (b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first hours that tickets were on sale. (c) For t 9, what is the fewest number of times at which L ( t) your answer. ʹ must equal? Give a reason for (d) The rate at which tickets were sold for t 9 is modeled by ( ) / rt = 55te t tickets per hour. Based on the model, how many tickets were sold by 3 P.. ( t = 3), to the nearest whole number. Page 8 of 9

30 Calculus aimus WS 8.: Integral as Net Change. AP-- t (minutes) Ht () (degrees Celsius) As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for t, H t is measured in degrees Celsius. Values of where time t is measured in minutes and temperature ( ) H( t ) at selected values of time t are shown in the table above. (a) Use the data in the table to approimate the rate at which the temperature of the tea is changing at time t = 3.5. Show the computations that lead to your answer. (b) Using correct units, eplain the meaning of ( ) H t dt in the contet of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate ( ) H t dt. (c) Evaluate Hʹ ( ) problem. t dt. Using correct units, eplain the meaning of the epression in the contet of this (d) At time t =, biscuits with temperature o C were removed from an oven. The temperature of the biscuits at time t is modeled by a differentiable function B for which it is known that t Bʹ t = 3.8e. Using the given models, at time t =, how much cooler are the biscuits than ( ).73 the tea? Page 9 of 9