APPLICATIONS OF

6.4 EXERCISES

1. How much work is done in lifting a 40-kg sandbag to a height 9. Suppose that 2 J of work is needed to stretch a spring from its of 1.5 m?

natural length of 30 cm to a length of 42 cm. 2. Find the work done if a constant force of 100 lb is used to pull

(a) How much work is needed to stretch the spring from 35 cm a cart a distance of 200 ft.

to 40 cm? (b) How far beyond its natural length will a force of 30 N keep 3. A particle is moved along the -axis by a force that measures x

the spring stretched?

10 兾共1 ⫹ x兲 2 pounds at a point feet from the origin. Find the x work done in moving the particle from the origin to a distance 10. If the work required to stretch a spring 1 ft beyond its natural

of 9 ft. length is 12 ft-lb, how much work is needed to stretch it 9 in. beyond its natural length?

4. When a particle is located a distance meters from the origin, x a force of 11. cos 共␲x兾3兲 newtons acts on it. How much work is

A spring has natural length 20 cm. Compare the work W 1

done in moving the particle from x苷1 to

? Interpret x苷2 done in stretching the spring from 20 cm to 30 cm with the your answer by considering the work done from

to work W 2 x苷1 done in stretching it from 30 cm to 40 cm. How are and from

W 2 and related? W x 苷 1.5 1 x 苷 1.5 x苷2

to

Shown is the graph of a force function (in newtons) that 12. If 6 J of work is needed to stretch a spring from 10 cm to increases to its maximum value and then remains constant.

12 cm and another 10 J is needed to stretch it from 12 cm How much work is done by the force in moving an object a

to 14 cm, what is the natural length of the spring? distance of 8 m?

13 – 20 Show how to approximate the required work by a Riemann F

sum. Then express the work as an integral and evaluate it. (N)

0.5 lb 兾ft and hangs over the 30 20 edge of a building 120 ft high.

13. A heavy rope, 50 ft long, weighs

10 (a) How much work is done in pulling the rope to the top of

the building?

0 1 2345678 x (m) (b) How much work is done in pulling half the rope to the top

of the building?

6. The table shows values of a force function f 共x兲 , where is x

14. A chain lying on the ground is 10 m long and its mass is

80 kg. How much work is required to raise one end of the Rule to estimate the work done by the force in moving an

measured in meters and f 共x兲 in newtons. Use the Midpoint

chain to a height of 6 m?

object from x苷4 to x 苷 20 . 15. A cable that weighs 2 lb 兾ft is used to lift 800 lb of coal up a

x 4 6 8 10 12 14 16 18 20 mine shaft 500 ft deep. Find the work done. f 共x兲

5 5.8 7.0 8.8 9.6 8.2 6.7 5.2 4.1 16. A bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket

7. A force of 10 lb is required to hold a spring stretched 4 in. is filled with 40 lb of water and is pulled up at a rate of 2 ft 兾s , beyond its natural length. How much work is done in stretching

but water leaks out of a hole in the bucket at a rate of 0.2 lb 兾s . it from its natural length to 6 in. beyond its natural length?

Find the work done in pulling the bucket to the top of the well. 8. A spring has a natural length of 20 cm. If a 25-N force is

17. A leaky 10-kg bucket is lifted from the ground to a height of required to keep it stretched to a length of 30 cm, how much

12 m at a constant speed with a rope that weighs 0.8 kg 兾m . work is required to stretch it from 20 cm to 25 cm?

Initially the bucket contains 36 kg of water, but the water

CHAPTER 6 APPLICATIONS OF INTEGRATION

leaks at a constant rate and finishes draining just as the bucket 26. Solve Exercise 22 if the tank is half full of oil that has a den- reaches the 12 m level. How much work is done?

sity of 900 kg 兾m 3 .

18. A 10-ft chain weighs 25 lb and hangs from a ceiling. Find the 27. When gas expands in a cylinder with radius , the pressure at r work done in lifting the lower end of the chain to the ceiling

any given time is a function of the volume: so that it’s level with the upper end.

P 苷 P共V 兲 . The force exerted by the gas on the piston (see the figure) is the 19. An aquarium 2 m long, 1 m wide, and 1 m deep is full of

product of the pressure and the area: F 苷 ␲r 2 P . Show that the water. Find the work needed to pump half of the water out

work done by the gas when the volume expands from volume of the aquarium. (Use the fact that the density of water is

V 1 to volume V 2 is

1000 kg 兾m 3 .) V 2 P dV

W苷

A circular swimming pool has a diameter of 24 ft, the sides

are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs

62.5 lb 兾ft 3 .)

piston head

A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 23 and 24 use the fact that

28. In a steam engine the pressure and volume P V of steam satisfy water weighs 62.5 lb 兾ft . 3 the equation PV 1.4 苷k , where is a constant. (This is true for k

21. 22. adiabatic expansion, that is, expansion in which there is no heat 3m

transfer between the cylinder and its surroundings.) Use Exer- 2m

1m

cise 27 to calculate the work done by the engine during a cycle when the steam starts at a pressure of 160 lb 兾in and a volume 2

3 3m 3 of 100 in and expands to a volume of 800 in . 8m

3m

29. Newton’s Law of Gravitation states that two bodies with masses m 1 and m 2 attract each other with a force 23. 6 ft

where is the distance between the bodies and r G is the gravi-

8 ft

6 ft

tational constant. If one of the bodies is fixed, find the work

3 ft

needed to move the other from r苷a to r苷b .

10 ft

frustum of a cone 30. Use Newton’s Law of Gravitation to compute the work required to launch a 1000-kg satellite vertically to an orbit

1000 km high. You may assume that the earth’s mass is ; 25. Suppose that for the tank in Exercise 21 the pump breaks

5.98 ⫻ 10 24 kg and is concentrated at its center. Take the down after

6.37 ⫻ 10 6 m and depth of the water remaining in the tank?

4.7 ⫻ 10 5 J of work has been done. What is the

radius of the earth to be

G 苷 6.67 ⫻ 10 ⫺11 N⭈m 2 兾 kg 2 .

6.5 AVERAGE VALUE OF A FUNCTION

It is easy to calculate the average value of finitely many numbers , y 1 y 2 ,..., y n :

y 1 ⫹y 2 ⫹⭈⭈⭈⫹y n

y ave 苷

But how do we compute the average temperature during a day if infinitely many tempera-

5 ture readings are possible? Figure 1 shows the graph of a temperature function T 共t兲 , where

6 T ave

t is measured in hours and T in C, and a guess at the average temperature, ⬚ T ave .

0 12 18 24 t

In general, let’s try to compute the average value of a function y 苷 f 共x兲 , a艋x艋b . We start by dividing the interval 关a, b兴 into n equal subintervals, each with length

FIGURE 1

⌬x 苷 共b ⫺ a兲兾n . Then we choose points x * 1 ,..., x * n in successive subintervals and cal-

SECTION 6.5 AVERAGE VALUE OF A FUNCTION

culate the average of the numbers f 共x 1 *兲 ,..., f 共x *兲 n :

f 共x 1 *兲 ⫹ ⭈ ⭈ ⭈ ⫹ f 共x n *兲 n

(For example, if represents a temperature function and f n 苷 24 , this means that we take temperature readings every hour and then average them.) Since ⌬x 苷 共b ⫺ a兲兾n , we can write n 苷 共b ⫺ a兲兾⌬x and the average value becomes

b⫺a i苷1 兺 f 共x *兲 ⌬x

If we let increase, we would be computing the average value of a large number of closely n spaced values. (For example, we would be averaging temperature readings taken every minute or even every second.) The limiting value is

b⫺a i苷1 兺

共x *兲 ⌬x 苷 i

b⫺a y a

by the definition of a definite integral.

Therefore we define the average value of f on the interval 关a, b兴 as

N For a positive function, we can think of this definition as saying

b⫺a y a

width 苷 average height

V EXAMPLE 1 Find the average value of the function

f 2 共x兲 苷 1 ⫹ x on the interval 关⫺1, 2兴 .

SOLUTION With and a 苷 ⫺1 b苷2 we have

3 b⫺a 苷2 2⫺ 冋 3 册 ⫺1

y a f 2 共x兲 dx 苷 x y ⫺1 共1 ⫹ x 兲 dx 苷 x⫹

f ave 苷

If T 共t兲 is the temperature at time , we might wonder if there is a specific time when the t temperature is the same as the average temperature. For the temperature function graphed in Figure 1, we see that there are two such times––just before noon and just before mid-

night. In general, is there a number at which the value of a function is exactly equal to c f the average value of the function, that is, f 共c兲 苷 f ave ? The following theorem says that this

is true for continuous functions.

THE MEAN VALUE THEOREM FOR INTEGRALS If is continuous on f 关a, b兴 , then there

exists a number in c 关a, b兴 such that

y a f b⫺a 共x兲 dx

f 共c兲 苷 f ave 苷

that is, b y

a f 共x兲 dx 苷 f 共c兲共b ⫺ a兲

CHAPTER 6 APPLICATIONS OF INTEGRATION

The Mean Value Theorem for Integrals is a consequence of the Mean Value Theorem for derivatives and the Fundamental Theorem of Calculus. The proof is outlined in Exer- cise 23.

The geometric interpretation of the Mean Value Theorem for Integrals is that, for posi- tive functions , there is a number such that the rectangle with base f c 关a, b兴 and height f 共c兲 has the same area as the region under the graph of from to . (See Figure 2 and the f a b

more picturesque interpretation in the margin note.)

y y=ƒ

N You can always chop off the top of a (two- dimensional) mountain at a certain height and use it to fill in the valleys so that the mountaintop becomes completely flat.

f(c)=f ave

FIGURE 2 0 a c b x

V EXAMPLE 2 Since f 共x兲 苷 1 ⫹ x 2 is continuous on the interval 关⫺1, 2兴 , the Mean Value Theorem for Integrals says there is a number in c 关⫺1, 2兴 such that

y (2, 5)

y ⫺1

2 共1 ⫹ x 2

兲 dx 苷 f 共c兲关2 ⫺ 共⫺1兲兴

y=1+≈

In this particular case we can find explicitly. From Example 1 we know that c f ave 苷2 ,

so the value of c satisfies

So in this case there happen to be two numbers c 苷 ⫾1 in the interval 关⫺1, 2兴 that work

_1 0 1 2 x

in the Mean Value Theorem for Integrals.

FIGURE 3

Examples 1 and 2 are illustrated by Figure 3.

V EXAMPLE 3 Show that the average velocity of a car over a time interval 关t 1 ,t 2 兴 is the

same as the average of its velocities during the trip.

SOLUTION If s 共t兲 is the displacement of the car at time , then, by definition, the average t velocity of the car over the interval is

On the other hand, the average value of the velocity function on the interval is

ave 苷

v 共t兲 dt 苷

t 2 ⫺t 1 y t 1 t 2 ⫺t 1 y t 1

s⬘ 共t兲 dt

关s共t 2 兲 ⫺ s共t 1 兲兴

(by the Net Change Theorem)

t 2 ⫺t 1

s 共t 2 兲 ⫺ s共t 1 兲

t 2 ⫺t 1 苷 average velocity

SECTION 6.5 AVERAGE VALUE OF A FUNCTION

6.5 EXERCISES

1– 8 Find the average value of the function on the given interval. 17. In a certain city the temperature (in F) hours after 9 ⬚ t AM 1. f 共x兲 苷 4x ⫺ x 2 , 关0, 4兴

2. f 共x兲 苷 sin 4 x, 关⫺␲, ␲兴 was modeled by the function

3. t共x兲 苷 s 3 x , 关1, 8兴 ␲t 4. t共x兲 苷 x 2 s1 ⫹ x 3 , 关0, 2兴 T 共t兲 苷 50 ⫹ 14 sin 12 5. f 共t兲 苷 te ⫺t 2 , 关0, 5兴 Find the average temperature during the period from 9 AM

6. f 共␪兲 苷 sec 2 共␪兾2兲,

to 9 PM .

7. h 共x兲 苷 cos 4 x sin x, 关0, ␲兴 18. (a) A cup of coffee has temperature 95 C and takes 30 min- ⬚ 8. h 共u兲 苷 共3 ⫺ 2u兲 ⫺1 ,

utes to cool to 61 C in a room with temperature 20 C. ⬚ 关⫺1, 1兴 ⬚ Use Newton’s Law of Cooling (Section 3.8) to show that the temperature of the coffee after minutes is t

共t兲 苷 20 ⫹ 75e ⫺kt (b) Find such c that f ave 苷 f 共c兲 .

(a) Find the average value of on the given interval. f T

where . k ⬇ 0.02 (c) Sketch the graph of and a rectangle whose area is the same f (b) What is the average temperature of the coffee during the

as the area under the graph of . f first half hour?

9. f 共x兲 苷 共x ⫺ 3兲 2 , 关2, 5兴 19. The linear density in a rod 8 m long is 12 兾 sx ⫹ 1 kg 兾m , 10. f 共x兲 苷 sx , 关0, 4兴

where is measured in meters from one end of the rod. Find x the average density of the rod.

; 11. f 共x兲 苷 2 sin x ⫺ sin 2x , 关0, ␲兴

If a freely falling body starts from rest, then its displacement ;

12. 共x兲 苷 2x兾共1 ⫹ x 2

f 兲 2 , 关0, 2兴

is given by

2 tt 2 . Let the velocity after a time T be v T . Show that if we compute the average of the velocities with

s苷 1

If is continuous and f x 3 1 f 共x兲 dx 苷 8 , show that takes on f respect to we get t ave 苷 2 T , but if we compute the average

the value 4 at least once on the interval 关1, 3兴 . ave 苷 3 v T . 14. Find the numbers such that the average value of b 21. Use the result of Exercise 79 in Section 5.5 to compute the

of the velocities with respect to we get s v

f 共x兲 苷 2 ⫹ 6x ⫺ 3x 2 on the interval 关0, b兴 is equal to 3.

average volume of inhaled air in the lungs in one respiratory

cycle.

The table gives values of a continuous function. Use the Mid-

point Rule to estimate the average value of on f 关20, 50兴 .

22. The velocity of blood that flows in a blood vessel with v radius and length at a distance from the central axis is R l r

16. The velocity graph of an accelerating car is shown. where is the pressure difference between the ends of the P (a) Estimate the average velocity of the car during the first

vessel and is the viscosity of the blood (see Example 7 in ␩ 12 seconds.

Section 3.7). Find the average velocity (with respect to ) r (b) At what time was the instantaneous velocity equal to the

over the interval 0艋r艋R . Compare the average velocity average velocity?

with the maximum velocity.

√ (km/h) 23. Prove the Mean Value Theorem for Integrals by applying the 60

Mean Value Theorem for derivatives (see Section 4.2) to the

function . F 共x兲 苷 x x a f 共t兲 dt

40 24. If f ave 关a, b兴 denotes the average value of on the interval f

20 关a, b兴 and , show a ⬍c⬍b that

c ⫺a

b ⫺c

0 4 8 12 t (seconds)

f ave 关a, b兴 苷

⫺a ave f 关a, c兴 ⫹ ave f 关c, b兴 b b ⫺a

CHAPTER 6 APPLICATIONS OF INTEGRATION

APPLIED

CAS

WHERE TO SIT AT THE MOVIES

PROJECT A movie theater has a screen that is positioned 10 ft off the floor and is 25 ft high. The first row

of seats is placed 9 ft from the screen and the rows are set 3 ft apart. The floor of the seating area is inclined at an angle of ␣ 苷 20⬚ above the horizontal and the distance up the incline that you sit is . The theater has 21 rows of seats, so x 0 艋 x 艋 60 . Suppose you decide that the best place to sit is in the row where the angle subtended by the screen at your eyes is a maximum. Let’s also ␪ suppose that your eyes are 4 ft above the floor, as shown in the figure. (In Exercise 70 in Sec- tion 4.7 we looked at a simpler version of this problem, where the floor is horizontal, but this

25 ft project involves a more complicated situation and requires technology.)

1. Show that

冉 2ab 冊

b 2 苷 共9 ⫹ x cos ␣兲 2 ⫹ 共x sin ␣ ⫺ 6兲 9 ft 2 2. Use a graph of as a function of to estimate the value of that maximizes . In which row ␪ x x ␪ should you sit? What is the viewing angle in this row? ␪ 3. Use your computer algebra system to differentiate and find a numerical value for the root ␪

and

of the equation d ␪ 兾dx 苷 0 . Does this value confirm your result in Problem 2? 4. Use the graph of to estimate the average value of on the interval ␪ ␪ 0 艋 x 艋 60 . Then use your CAS to compute the average value. Compare with the maximum and minimum values

of . ␪