FURTHER APPLICATIONS OF INTEGRATION

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8 FURTHER APPLICATIONS OF INTEGRATION

  For instance, we willinvestigate the center of gravity of a plate, the force exerted by water pressure on a dam, the flow of blood from the human heart, and the average time spent on hold during acustomer support telephone call. (We can use the distance formula to find the distance of Figure 2. between the endpoints of each segment.) We are going to define the length of a general curve by first approximating it by a polygon and then taking a limit as the number of seg-ments of the polygon is increased.

8.1 What do we mean by the length of a curve? We might think of fitting a piece of string to

  We need a precisedefinition for the length of an arc of a curve, in the same spirit as the definitions we devel- oped for the concepts of area and volume. (We can use the distance formula to find the distance of Figure 2. between the endpoints of each segment.) We are going to define the length of a general curve by first approximating it by a polygon and then taking a limit as the number of seg-ments of the polygon is increased.

1 P i⫺ P i

  L 苷 lim 1n l ⬁ i苷1 Pi-1 Pi Notice that the procedure for defining arc length is very similar to the procedure we used for defining area and volume: We divided the curve into a large number of small parts. n l ⬁ The definition of arc length given by Equation 1 is not very convenient for compu- Pi-1 tational purposes, but we can derive an integral formula for in the case where has a L f continuous derivative.

1 P i ⌬y i f ⬘ x i * ⌬x

  苷 s ⌬x 苷 s ⌬x 2 2 2i i (since ) ⌬x ⌬x ⬎ Therefore, by Definition 1, n n ⌬x P i⫺ 1 P i iL 苷 lim 苷 lim s1 ⫹ f ⬘ x n n l l ⬁ ⬁i苷1 i苷1 We recognize this expression as being equal to b 2 y a dx s1 ⫹ f ⬘ x by the definition of a definite integral. This integral exists because the function 2 is continuous.

THE ARC LENGTH FORMULA

f ⬘ a, b the curve , , is

2 If is continuous on , then the length of

  a 艋 x 艋 b y 苷 f xb 2 dx L 苷 s1 ⫹ f ⬘ xy a If we use Leibniz notation for derivatives, we can write the arc length formula as follows:b 2dy 3 1 ⫹ dx L 苷y a dx 2 3 EXAMPLE 1 Find the length of the arc of the semicubical parabola between the y 苷 x y points and . (See Figure 5.)1, 1 4, 8 (4, 8)SOLUTION For the top half of the curve we have dy 3 3 2 1 2x ¥=˛ y 苷 x 苷 2dx and so the arc length formula gives (1, 1) x 2 4 4dy 9 1 ⫹ 1 ⫹ dx x L 苷 dx 苷 s 4y 1y 1 dxF I G U R E 5 9 9 13 If we substitute , then dx .

SECTION 8.1 ARC LENGTH |||| 527

As a check on our answer to Example 1, notice Therefore from Figure 5 that the arc length ought to be 10 10 4 4 2 2 3slightly larger than the distance from 1, 1 tou su ] 13 3y 4 134, 8 , which is 4 L 苷 9 du 苷 9 ⴢ 3 2 8 2 13 17.615773 3s58 ⫺ ⫺ 10 80 13 M [ ( ) ] ( s10 s13 ) 苷 27 4 苷 27 According to our calculation in Example 1, wehave If a curve has the equation , c 艋 y 艋 d , and is continuous, then by inter-1 x 苷 t y t⬘ y⫺ 80 13 7.633705L 苷 27 ( s10 s13 ) changing the roles of and in Formula 2 or Equation 3, we obtain the following formula x y Sure enough, this is a bit greater than the length for its length: of the line segment.d d 2dx 2 4 1 ⫹ dy L 苷 s1 ⫹ t⬘ y dy 苷 y cy c dy 2 V EXAMPLE 2 Find the length of the arc of the parabola y from 0, 0 to 1, 1 . 苷 x

2 SOLUTION

  Notice that each time we 32 1.478double the number of sides of the polygon, we x 164 1.479get closer to the exact length, which is ⫹s5 ln ( s5 2 ) ⫹ 1.478943L 苷 F I G U R E 6 2 4 528 ||||CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION Because of the presence of the square root sign in Formulas 2 and 4, the calculation of an arc length often leads to an integral that is very difficult or even impossible to evaluateexplicitly. V EXAMPLE 3 (a) Set up an integral for the length of the arc of the hyperbola from the xy 苷 1 1 point 1, 1 to the point 2, .

1 Checking the value of the definite integral

  ⌬x with a more accurate approximation produced by f 1 ⫹ 4 f 1.1 ⫹ 2 f 1.2 ⫹ 4 f 1.3 ⫹ ⭈ ⭈ ⭈ ⫹ 2 f 1.8 ⫹ 4 f 1.9 ⫹ f 2 a computer algebra system, we see that the 3 approximation using Simpson’s Rule is accurate to four decimal places. Thus if a smooth curve C has theequation , a 艋 x 艋 b , let s x be the distance along C from the initial point y 苷 f x P a, f a to the point Q x, f x .

SECTION 8.1 ARC LENGTH |||| 529

  Equation 6 shows that the rate of change of with respect to is always at least 1 and is s x equal to 1 when f ⬘ x , the slope of the curve, is 0. The differential of arc length is 2dy 7 1 ⫹ dx ds 苷 dx and this equation is sometimes written in the symmetric form 2 2 28 ⫹ ds dy苷 dx y The geometric interpretation of Equation 8 is shown in Figure 7.

2 SOLUTION ⫺

  Figure 9 shows thex 1s x 1graph of this arc length function. Why is y=≈- ln x 8xnegative when is less than ?

8.1 E X E R C I S E S

  Use the arc length formula (3) to find the length of the curvey 苷 ln 1 ⫺ x , 0 艋 x 艋2 ⫺, 1 艋 x 艋 3 . y , 0 艋 x 艋 2 , y ⬎ 苷 4 x ⫹ 45 x 1 9.⫹ y 苷 , 1 艋 x 艋 2323 – 26 Use Simpson’s Rule with n 苷 10 to estimate the arc 6 10x4length of the curve.

SECTION 8.1 ARC LENGTH |||| 531

  0 艋 x 艋 2␲ 4 ⫺ x y 苷 x s3 0 艋 x 艋 4 n 苷 1 2 4 y 苷 x ⫹ sin x CAS y 苷 ln x x2 2, ln 2 1, 0 3 CAS y 苷 x4 苷 1 31.1, 1 0, 0 3 ⫹ y2 3 37. Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve that liesbetween the points and .

8.2 A surface of revolution is formed when a curve is rotated about a line. Such a surface is the lateral boundary of a solid of revolution of the type discussed in Sections 6.2 and 6.3

  The lateral surface area of a circular cylinder with radius and height is taken to be because we can imagine cutting the cylin- r h A 苷 2␲rhcut h der and unrolling it (as in Figure 1) to obtain a rectangle with dimensions 2␲r and .h Likewise, we can take a circular cone with base radius and slant height , cut it along r lr the dashed line in Figure 2, and flatten it to form a sector of a circle with radius and cen- l tral angle . We know that, in general, the area of a sector of a circle with radius␪ 苷 2␲r l 1 2 ␪ ␪ and angle is (see Exercise 35 in Section 7.3) and so in this case the area is l l 2h 2␲r 1 1 2 22πrl l A 苷 2 ␪ 苷 2 苷 ␲rll F I G U R E 1 Therefore we define the lateral surface area of a cone to be .

2 A 苷 ␲rl

  yy (x, y)y x (x, y) xcircumference circumference =2πy =2πx xF I G U R E 5 (a) Rotation about x-axis: S= (b) Rotation about y-axis: j  2πy ds S=j 2πx ds 2 2 2 V EXAMPLE 1 The curve , ⫺ 1 艋 x 艋 1 , is an arc of the circle x ⫹ y .y 苷 s4 ⫺ x 苷 4Find the area of the surface obtained by rotating this arc about the -axis. (The surface is x a portion of a sphere of radius 2. See Figure 6.) SOLUTION We have ydy ⫺x 1 兾2 2 ⫺ 1 共4 ⫺ x 兲共⫺2x兲 苷 苷 2 2dx s4 ⫺ x and so, by Formula 5, the surface area is 2 1 dy 2␲ y 1 ⫹ dx 1 x S 苷y ⫺ 1dx 冑 冉 冊 2 1 x 2 y 1 ⫹ dx s4 ⫺ x苷 2␲ 2 ⫺ 1 4 ⫺ x 冑 1 2 2 y 2 ⫺ dx s4 ⫺ x苷 2␲ 1 s4 ⫺ xN F I G U R E 6 1 Figure 6 shows the portion of the sphereM 苷 4␲ 1 dx 苷 4␲共2兲 苷 8␲ whose surface area is computed in Example 1.

2 Substituting , we have . Remembering to change the limits of

  x 苷 ln yx x dy and y 苷 e 苷 e dx 0 艋 x 艋 1 y 苷 ln x xn 苷 10 y 苷 e ⫺x2 0 艋 x 艋 ␲ 兾3 y 苷 sec x 1 艋 x 艋 2 y 苷 x ⫹ sx 1 艋 x 艋 3 1– 4 y 苷 1兾x xCAS Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about (a) the -axis and(b) the -axis. , 21– 22 Use either a CAS or a table of integrals to find the exact area of the surface obtained by rotating the given curve about the 2 冉dy dx 兾4 1 sec 3 ␪ d␪ u 苷 e x 苷 2␲ y e s1 ⫹ u 1 ⫹ 2 duS 苷 y 1 2␲ y 冑 8.2 N Or use Formula 21 in the Table of Integrals.

26. CAS

  If the dish is to have a 10-ft diameter and a ybetween two parallel planes is , where is the diam- d S 苷 ␲ dh maximum depth of 2 ft, find the value of and the surface a eter of the sphere and is the distance between the planes. 31.a 艋 x 艋 b c S If the curve y 苷 f 共x兲 , , is rotated about the If is a positive constant, define t共x兲 苷 f 共x兲 ⫹ c and let t f 共x兲 艋 c horizontal line y 苷 c , where , find a formula for the be the corresponding surface area generated by the curve a 艋 x 艋 b S S L area of the resulting surface.

2 P 苷 ␳td 苷 ␦d , where is the weight

  Since this is a small unit, the kilopascal (kPa) is often苷 1 density (as opposed to , which is the mass ␳ 3 used. ␦ 苷 62.5 lb兾ft 3 2 ⫻ 9.8 m 兾s ⫻ 2 mP 苷 ␳td 苷 1000 kg兾m 苷 19,600 Pa 苷 19.6 kPaAn important principle of fluid pressure is the experimentally verified fact that at any point in a liquid the pressure is the same in all directions.

1 P 苷 ␳td 苷 ␦d

  Find the force on the dam due to hydrostatic pressure if the water level is 4 m from the top of the dam. The depth of the water is 16 m, so we divide the interval 关0, 16兴 into sub- 30 m intervals of equal length with endpoints x i and we choose x i i , x i 兴 .

10 If ⌬x is small, then the pressure P i on the i th strip is almost constant and we can use

  From the equation of the circle, we see that the length of the strip isand so its area is The pressure on this strip is approximately and so the force on the strip is approximatelyThe total force is obtained by adding the forces on all the strips and taking the limit: The second integral is 0 because the integrand is an odd function (see Theorem 5.5.7). Thus 2 3 ⫺ 3 共7 ⫺ y兲 s9 ⫺ y 2 dyF 苷 lim n l ⬁ 兺 ni苷1 s9 ⫺ y dy 苷 875 ⴢ 2 s9 ⫺ y y s9 ⫺ y 3 3 ⫺ y dy ⫺ 125 2 3 1 3 ⫺ y y 2y 苷 3 sin ␪ ␲共3兲 2 苷 125 ⴢ 7 542 ||||CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION MOMENTS AND CENTERS OF MASSOur main objective here is to find the point on which a thin plate of any given shape bal- P P ances horizontally as in Figure 5.

2 F I G U R E 5

  , x n on the -axis, it x can be shown similarly that the center of mass of the 1 2 system is located at n n m i x i m i x i 兺 兺i苷1 i苷1 4x 苷 n 苷 m m i 兺i苷1 where 冘 m i is the total mass of the system, and the sum of the individual moments m 苷 n m i x i M 苷兺i苷1 is called the moment of the system about the origin. By analogy with 1 1 2 2‹ m¡m£ the one-dimensional case, we define the moment of the system about the y-axis to be › y£n xfi 5 M y m i x i 苷 兺m™i苷1 ¤ and the moment of the system about the x-axis as F I G U R E 8n 6 M x m i y i 苷 兺i苷1 Then M y measures the tendency of the system to rotate about the -axis and y M x measures the tendency to rotate about the -axis.

3 Since , we use Equations 7 to obtain

  But the mass of the plate is the product of its density and its area: mx 苷 M my 苷 M b f m 苷 ␳A 苷 ␳ 共x兲 dxy a and so b b x f x f␳ 共x兲 dx 共x兲 dx y y a aM y x 苷 苷 苷 b b m f f␳ 共x兲 dx 共x兲 dx y y b b a a 1 1 2 2 dx dx␳ 关 f 共x兲兴 关 f 共x兲兴 2 2y y a a M x y 苷 苷 苷 b b m f f␳ 共x兲 dx 共x兲 dx y y a a Notice the cancellation of the ’s. SECTION 8.3 APPLICATIONS TO PHYSICS AND ENGINEERING |||| 545 ᏾In summary, the center of mass of the plate (or the centroid of ) is located at the point , where共x, y兲 b b 1 1 1 2 8 x f dx x 苷 共x兲 dx y 苷 关 f 共x兲兴 2y y a a A A EXAMPLE 4 Find the center of mass of a semicircular plate of radius .

1 If the region lies between two curves and , where , as

f 共x兲 艌 y 苷 f 共x兲 y 苷 t共x兲 t共x兲关 C ”x , f(x )+g(x )兴’i i i i 2 2 illustrated in Figure 13, then the same sort of argument that led to Formulas 8 can be used᏾ y=ƒ to show that the centroid of is , where共x, y兲 ᏾b 1 9y=© x关 f 共x兲 ⫺ x 苷 t共x兲兴 dx y aAx a bx xi ib 1 1 2 2 ⫺兵关 f 共x兲兴 关 其 dx y 苷 2 t共x兲兴F I G U R E 1 3y a A (See Exercise 47.) EXAMPLE 6 Find the centroid of the region bounded by the line and the y 苷 x 2 parabola .y 苷 x 2 SOLUTIONy The region is sketched in Figure 14. We take , , , andf 共x兲 苷 x a 苷 0 t共x兲 苷 xin Formulas 9. First we note that the area of the region is b 苷 1(1, 1) y=x 1 1 2 2 3”  ,   ’ 1 1 x x 2 5 2 ⫺共x ⫺ x 兲 A 苷 dx 苷 苷y 2 3 6 册y=≈ x Therefore F I G U R E 1 4 1 1 1 1 2 x x关 f 共x兲 ⫺ 共x ⫺ x 兲 dx x 苷 1 t共x兲兴 dx 苷 y yA 6 1 3 4 1 1 x x 2 3 ⫺ x ⫺共x 苷 6 兲 dx 苷 6 苷 y 3 4 2 冋 册 1 1 1 1 1 1 2 2 2 4 ⫺ ⫺ x兵关 f 共x兲兴 关 共x 兲 dx y 苷 2 t共x兲兴 其 dx 苷 1 2y y A 6 1 3 5 2 x x ⫺苷 3 苷 3 5 5 冋 册 1

2 The centroid is ,

  2y 苷 sin x , y 苷 cos x , x 苷 0 , x 苷 ␲兾43 2 32.x 1y 苷 x , , y 苷 0 ⫹ y 苷 22 1 33.x 苷 5 ⫺ y , x 苷 0x _2 _1 1 2 x_2 1 3 _134 –35 M M Calculate the moments x and y and the center of mass of a lamina with the given density and shape. If is the -coordinate of the centroid of the region that lies x x 4 under the graph of a continuous function , where f a 艋 x 艋 b , show that 2b b f x dx cx ⫹ d f x dx 苷 cx ⫹ d y y a ax 2 4 6 844 – 46 Use the Theorem of Pappus to find the volume of the x given solid.

37. Find the centroid of the region bounded by the curves y 苷 2

  The solid obtained by rotating the triangle with vertices 38.x Use a graph to find approximate -coordinates of the points ;3 2, 3 , 2, 5 , and about 5, 4 the x -axis ⫺ x of intersection of the curves y 苷 x ⫹ ln x and y 苷 x . ᏾ Let be the region that lies between the curvesProve that the centroid of any triangle is located at the pointn y 苷 xand , 0 艋 x 艋 1 , where m and are integers with n of intersection of the medians.

2. A A

  What does your result from Problem 1 say about the areas 1 and 2 shown in the figure? 4.h Based on your own measurements and observations, suggest a value for and an equation for x 苷 f 共 y兲 and calculate the amount of coffee that each cup holds.

8.4 In this section we consider some applications of integration to economics (consumer sur- plus) and biology (blood flow, cardiac output). Others are described in the exercises

  P 苷 p共X兲 We divide the interval 关0, X兴 into subintervals, each of length n , and let⌬x 苷 X兾n (X, P) x i i be the right endpoint of the th subinterval, as in Figure 2. If, after the first i x i⫺ 1 P units were sold, a total of only x i units had been available and the price per unit had been set at p 共x i 兲 dollars, then the additional ⌬x units could have been sold (but no more).

SECTION 8.4 APPLICATIONS TO ECONOMICS AND BIOLOGY |||| 551

  p Considering similar groups of willing consumers for each of the subintervals and adding the savings, we get the total savings:n关p共x i 兲 ⫺ P兴 ⌬x 兺i苷1 (This sum corresponds to the area enclosed by the rectangles in Figure 2.) If we let , n l ⬁(X, P) this Riemann sum approaches the integral PX 1 关p共x兲 ⫺ P兴 dx yx ⁄ x i X which economists call the consumer surplus for the commodity. Figure 3 P X shows the interpretation of the consumer surplus as the area under the demand curve and p=p(x) above the line .p 苷 P V EXAMPLE 1 The demand for a product, in dollars, is consumer surplus 2p 苷 1200 ⫺ 0.2x ⫺ 0.0001x P p=P Find the consumer surplus when the sales level is 500.

2 F 苷

  If we divide 关0, T 兴 into subintervals of equal length ⌬t , then the amount of dye that flows vein past the measuring point during the subinterval from i to i is approximately t 苷 t ⫺ 1 t 苷 tF I G U R E 6i F ⌬t concentration volume 苷 c t where is the rate of flow that we are trying to determine. Thus the total amount of dye is approximatelyand, letting , we find that the amount of dye isThus the cardiac output is given by where the amount of dye is known and the integral can be approximated from the con-centration readings.

3 A 苷 F

SECTION 8.4 APPLICATIONS TO ECONOMICS AND BIOLOGY |||| 553

  Thus the total amount of dye is approximatelyand, letting , we find that the amount of dye isThus the cardiac output is given by where the amount of dye is known and the integral can be approximated from the con-centration readings. If the revenue from the sale of the first 1000 units is $12,400, find the revenue from the sale of thefirst 5000 units.

2. The marginal revenue from the sale of units of a product

  16 5.1 4 2.3 14 共t兲 ; t f 共t兲 苷 9000s1 ⫹ 2t t st f ⬘ 共t兲 f 共t兲 t 10.p 苷 800,000e ⫺x 兾5000x ⫹ 20,000 p 苷 20 ⫹110 xp 苷 50 ⫺120 xp 苷 200 ⫹ 0.2x 3 / 2 X 苷 10p S 共x兲 苷 3 ⫹ 0.01x2yX 关P ⫺ p S 共x兲兴 dx 554 ||||CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATIONt t 12 3.9 A company modeled the demand curve for its product(in dollars) by the equationUse a graph to estimate the sales level when the selling price is $16. Use Poiseuille’sLaw to show that if and are normal values of the radius and pressure in an artery and the constricted values are and , then for the flux to remain constant, and are related by the equationDeduce that if the radius of an artery is reduced to three- fourths of its former value, then the pressure is more thantripled.

11. If the amount of capital that a company has at time is , then the derivative, , is called the net investment flow

  For a given commodity and pure competition, the number of P P , find the producer surplus when the selling price is $400. If a supply curve is modeled by the equation 2200 ⫹ 10e 0.8t argument similar to that for consumer surplus shows that thesurplus is given by the integralCalculate the producer surplus for the supply function at the sales level .

12. If revenue flows into a company at a rate of

13. Pareto’s Law of Income

  Ax ⫺k dx x 苷 b x 苷 a 14 4 6 2 4 10 2 8 12 N 苷x b a 6c 共 t兲 states that the number of people with incomes between and is , whereand are constants with and . The average income of these people is Calculate .

SECTION 8.5 PROBABILITY |||| 555

PROB ABILITY

8.5 Calculus plays a role in the analysis of random behavior. Suppose we consider the choles-

  We might want to know the probability that a blood cholesterol level is greater than 250, or the probability that the height of an adult female is between 60 and70 inches, or the probability that the battery we are buying lasts between 100 and 200 hours. If X represents the lifetime of that type of battery, we denote this last probability asfollows: P 共100 艋 X 艋 200兲 According to the frequency interpretation of probability, this number is the long-run pro- portion of all batteries of the specified type whose lifetimes are between 100 and 200hours.

1 P f 共x兲 dx

  共a 艋 X 艋 b兲 苷 y a For example, Figure 1 shows the graph of a model for the probability density function f for a random variable X defined to be the height in inches of an adult female in the United States (according to data from the National Health Survey). If is the probability density function and you call at time f 2 , then, from Definition 1, f 共t兲 dt represents the probability that an agent answers t 苷 0x 5 within the first two minutes and f 共t兲 dt is the probability that your call is answered x 4 during the fifth minute.

SECTION 8.5 PROBABILITY |||| 557

  Let f be the corresponding density function,共t兲 y=f(t) where t is measured in minutes, and think of a sample of N people who have called this company. (Think of ⌬t as lasting a minute, or half a minute, or 10 sec- 1 2 60 onds, or even a second.) The probability that somebody’s call gets answered during the time period from t i to is the area under the curve t i from t i to , which is t i ⫺ 1 y 苷 f 共t兲 ⫺ 1tt t t approximately equal to f i .

i- 1 i 共t 兲 ⌬t

  ti ure 3, where is the midpoint of the interval.) t iSince the long-run proportion of calls that get answered in the time period from t i to F I G U R E 3⫺ 1t i is f i , we expect that, out of our sample of N callers, the number whose call was 共t 兲 ⌬t answered in that time period is approximately N f i and the time that each waited is共t 兲 ⌬t about . 冉 冊 The mean is , so we can rewrite the probability density function as␮ 苷 1兾c if t ⬍ 0 fM 共t兲 苷 兾␮ ⫺1 ⫺te if t 艌 0 再 ␮ V EXAMPLE 4 Suppose the average waiting time for a customer’s call to be answered by a company representative is five minutes.(a) Find the probability that a call is answered during the first minute.(b) Find the probability that a customer waits more than five minutes to be answered.

SECTION 8.5 PROBABILITY |||| 559

  In general, the median of a probability density func- tion is the number m such that ⬁ 1 f共x兲 dx 苷 2y m This means that half the area under the graph of lies to the right of m. ␴ graphs of members of the family in Figure 5, we see that for small values of the values␴ of X are clustered about the mean, whereas for larger values of the values of X are more ␴ spread out.

0.01 V EXAMPLE 5

  P 共1 艋 X 艋 2兲 f x ⬍ 0 f 共x兲 苷 0 x 艌 共x兲 苷 xe ⫺x x ⬍ 0 f P ( X ⬍ 2兲 f xf 共x兲 苷 0 0 艋 x 艋 4 f 共x兲 苷3 64 x s16 ⫺ x2 t f 共t兲y ⬁ 25,000 f 共x兲 dx y 40,000 30,000 f 共x兲 dx x f 共x兲 1. E X E R C I S E S 共x兲 苷 0 0 艋 x 艋 1 1 (It’s quite safe to say that people with an IQ over 200 are extremely rare.) ThenTherefore about 0.4% of the population has an IQ over 140. The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2.5 min-utes.(a) Find the probability that a customer has to wait more than 4 minutes.(b) Find the probability that a customer is served within the first 2 minutes.(c) The manager wants to advertise that anybody who isn’t served within a certain number of minutes gets a free ham-burger.

9. Show that the median waiting time for a phone call to the com- pany described in Example 4 is about 3.5 minutes

  The integralgives the probability that the electron will be found within the sphere of radius meters centered at the nucleus.(a) Verify that is a probability density function.(b) Find . (d) Find the probability that the electron will be within the sphere of radius centered at the nucleus.(e) Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom.

14. Boxes are labeled as containing 500 g of cereal. The machine

0.2 P 共3 艋 X 艋 8兲 P 共X ⬍ 3兲 8

SECTION 8.5 PROBABILITY |||| 561

  The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2.5 min-utes.(a) Find the probability that a customer has to wait more than 4 minutes.(b) Find the probability that a customer is served within the first 2 minutes.(c) The manager wants to advertise that anybody who isn’t served within a certain number of minutes gets a free ham-burger. (d) Find the probability that the electron will be within the sphere of radius centered at the nucleus.(e) Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom.

8 C O N C E P T C H E C K 1

  Given a demand function , explain what is meant by the(b) Write an expression for the length of a smooth curve given consumer surplus when the amount of a commodity currently a 艋 x 艋 b X P by , y 苷 f 共x兲 . 艋 x 艋 b obtained by rotating the curve , a , about y 苷 f 共x兲 (b) Explain how the cardiac output can be measured by the dye the -axis.

3. Describe how we can find the hydrostatic force against a verti- 9

  of a female college student, where is measured in pounds.(a) What is the physical significance of the center of mass of a 130 共x兲 dx(a) What is the meaning of the integral f ? (b) Write an expression for the mean of this density function.(b) If the plate lies between and , where y 苷 f 共x兲 y 苷 0 (c) How can we find the median of this density function?艋 x 艋 b a , write expressions for the coordinates of the center of mass.

10. A trough is filled with water and its vertical ends have the Find the area of the resulting surface

  Show that the surface area of the spherical zone equals the surface area of the region that the two planes cut off on the cylinder.(d) The Torrid Zone is the region on the surface of the earth that is between the Tropic of of two bases Use mi for the radius of the earth.(c) A sphere of radius is inscribed in a right circular cylinder of radius . The total set of possibilities for the needle can be identified with the rectangular L region 0 艋 y 艋 L , 0 艋 ␪ 艋 ␲ , and the proportion of times that the needle intersects a line is the ratioarea under y 苷 h sin ␪ area of rectangle y This ratio is the probability that the needle intersects a line.

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