27.1 Finding ÒMassÓ When Density Varies 841
(b) Another trufße is made in a hemispherical mold with radius . Layers of different types of chocolate are poured into the mold, one at a time, and allowed to set. The number of calories per cubic millimeter varies with , where is the depth from the top of the mold. The calorie density is given by / calories/mm3. Write an integral that gives the number of calories in this hemispherical trufße.
Top of the mold
Hemispherical truffle mold
14. Liquidisbeingstoredinalargesphericaltankofradius2meters.Thetankiscompletely full and has been left standing for a long time. A mineral suspended in the liquid is setting. Its density at a depth of meters from the top is given by 5 milligrams per cubic meter. Determine the number of milligrams of the mineral contained in the tank.
15. A circular pond is 60 meters in diameter and has a bridge running along a diameter. At lunchtime people stand on the bridge and throw bread onto the water to feed the ducks. As a result, the density of ducks on the pond is given by a function , ducks per square meter, where is the distance from the bridge. How many ducks are on the pond? (We will assume that the bridge itself is very thin so we can ignore its width.)
Notice that we cannot really say that the ducks are continuously distributed on the pond. Ducks, after all, are discrete. We are making a continuous model of a discrete phenomenon.
16. Letbetheamountofwaterinapoolattime, measuredinhoursand measured
in gallons. 0 corresponds to noon. Water is ßowing in and out of the pool at a rate given by 30 cos . During what time interval between noon and 5:00 p.m.
0 5 is water ßowing out of the pool at a rate of 15 gallons an hour or more? How much water actually has left the pool in this time interval?
17. In the town of Lybonrehc there has been a nuclear reactor meltdown, which released radioactive iodine 131. Fortunately, the reactor has a containment building, which kept the iodine from being released into the air. The containment building is hemispherical with a radius of 100 feet. The density of iodine in the building was 6 105200 g/cubic feet, where is the height from the ßoor (in feet). (It ranges from 12 103 g/cubic feet at the ßoor to 6 103 g/cubic feet near the top.)
(a) Deriveanintegralthatgivestheamountofiodineinthebuilding.Youmustexplain your reasoning fully and clearly.
(b) Calculate the amount of iodine in the building.
842 CHAPTER 27 Applying the DeÞnite Integral: Slice and Conquer
18. A spherical star has a radius of 90,000 kilometers. The density of matter in the star
is given by ,- -13 2 kilograms per cubic kilometer, where - is the distance (in kilometers) from the starÕs center and K is a positive constant.
Write out (but do not evaluate) an expression for the total mass of the star. Your answer should contain the constant K.
19. A substance has been put in a centrifuge. We now have a cylindrical sample (radius 3 centimeters, height 4 centimeters) in which density varies with , the distance (in centimeters) from the central axis. If the density is given by , mg/cm3, write an integral that gives the total mass of the substance.
20. A very thin, lighted pole 10 feet tall is placed upright in a familyÕs backyard to attract insects to it (where they are electrocuted). At one moment, the density of these insects
is given by ,- -1 insects per cubic foot, where - measures the number of feet from the pole.
(a) How many insects are within 5 feet of the pole at a height of 10 feet or less? (b) How many insects are within 5 feet of the pole at a height of 10 feet or more?
21. A circus tent has cylindrical symmetry about its center pole. The height a distance of
feet from the center pole is given by 8 feet. What is the volume enclosed 1
by the tent of radius 4?
22. At the Three Aces pizzeria, the chef tosses lots of garlic on the pizza. The density of garlic varies with , the distance from the center of the pizza, and is given by
) 3 22 ounces per square inch of pizza.
If the pizza is 14 inches in diameter, and Three Aces cuts six slices from each pizza, how much garlic is on one slice of pizza? (Problem by Andrew Engelward)
23. (a) What is the present value of a single payment of $2000 three years in the future? Assume 5% interest compounded continuously.
(b) What is the present value of a continuous stream of income at the rate of $100,000 per year over the next 20 years? Assume 5% interest compounded continuously. By Òa continuous stream of incomeÓ we mean that we are modeling the situation by assuming that money is being generated continuously at a rate of $100,000 per year.
Begin by partitioning the time interval [0, 20] into equal pieces. Figure out the amount of money generated in the th interval and pull it back to the present. Summing these pull-backs should approximate the present value of the entire income stream.
27.2 Slicing to Find the Area Between Two Curves 843
27.2 SLICING TO FIND THE AREA BETWEEN TWO CURVES
Section27.1istheheartoftheÒslicingÓdiscussion.Itallowsustoapplywhatweknowabout integration to a very broad array of situations. In this section weÕll use the same approach to enlarge the type of area problems we can deal with.
EXAMPLE 27.5 Find an integral that gives the shaded area in Figure 27.10 the area bounded by " , " ), and the vertical lines and .
y = f(x)
y = g(x) x
x = a x = b
SOLUTION Chop the area into tall, thin ÒrectanglesÓ by partitioning the interval [, ] along the -axis into equal pieces as shown; each piece is of length and .5
a x b x0 x1 x2 x3 xn—1 xn
Notice that the height of each rectangle is of the form ), regardless of where the -axis is in relation to the graphs of and ). To Þnd any vertical distance, just subtract the bottom (lower) "-value from the top (higher) "-value. (See Figure 27.11.)
y = —3
—3 — (—7) = —3 + 7 = 4
y = —7
y = 6
y = 2
6 — 1 = 5 x
2 — (—3) = 2 + 3 = 5 x
y = —3 y = 1
the area of the th rectangle (height) (length) [)]
5 When we chop up the area into many pieces we get a slew of pseudo-rectangles. They donÕt have ßat tops and ßat bottoms, so they are not really rectangles. But each one can be closely approximated by a genuine rectangle, one end of which lies on the curve " and the other end on the curve " ). For the remainder of this section, for ease of discussion we will say that we chop regions up into rectangles when more precisely what we mean is that we chop the region into pseudo-rectangles.
844 CHAPTER 27 Applying the DeÞnite Integral: Slice and Conquer
the area of the region [)] 1
the area of the region lim [)]
y = f(x)
f(xi) — g(xi)
y = g(x)
y = f(x)
y = g(x)
x = a x = b
EXAMPLE 27.6 Find an integral that gives the shaded area in Figure 27.13, the area bounded by 0", -", and the horizontal lines " and " .
x = q(y)
x = r(y)
SOLUTION Chop the area horizontally into long, narrow rectangles by partitioning the interval [, ] along the "-axis into equal pieces as shown; each piece is of height " . Notice that the length of each rectangle is of the form 0"-", regardless of where the "-axis is in relation to the graphs of 0 and -. To Þnd any horizontal distance, we subtract the -value on the left (the smaller one) from the -value on the right (the larger one). (See Figure
27.2 Slicing to Find the Area Between Two Curves 845
yn d .
y3 }y y2
x = 1 x = 3 x = —1 x = 3 4
3—1 = 2 3— (—1) = 4
x = —5 x = —2
—2— (—5) = —2 + 5 = 3
the area of the th rectangle (length) (height) [0"-"] "
q(yi) — r(yi) x = q(y) x = r(y)
x = q(y)
cx = r(y)
the area of the region [0"-"] " 1
the area of the region lim [0"-"] "
EXERCISE 27.5 Find the area bounded above by " 4 and below by
(a) " 3, (b) " 5, (c) " 6.
Answers are given at the end of the section.
EXERCISE 27.6 Which of the following expressions gives the area bounded on the left by the "-axis, on the right by " ln , below by the -axis, and above by the line " 1? (There is more than one correct answer.)