Northfield Mount Hermon School Advanced Placement Calculus Problems in the Use of the Definite Integral Dick Peller Winter 999 PROBLEM. Oil is leaking from a tanker at the rate of R(t) = e -.t gallons per hour. How much oil has leaked out of the tanker after hours? R( t) dt = e.t dt 8,647gallons PROBLEM. The EPA was recently asked to investigate a spill of radioactive iodine. Measurements showed the ambient radiation levels at the site to be four times the maximum acceptable level, so the EPA ordered an evacuation of the surrounding neighborhood. It is known that the level of radiation from an iodine source decreases according to the formula R(t) = R e -.4t, where R(t) is the radiation level (in millirems/hour) a time t, and R is the initial radiation level, and t is time in hours. a) How long will it take the site to reach an acceptable level of radiation? b) How much total radiation (in millirems) will have been emitted by this time, assuming the maximum acceptable limit is.6 millirems/hour? a) e -.4t =.5, so t = 347 hours..4t b).4e dt = 45 millirems Page of 9
PROBLEM 3. Suppose the density of a circular oil slick on the surface of the water is given by ρ( r) = /. + r kg m a) Given that the slick extends from r= to r= meters, write a Riemann sum that approximates the total mass of oil. b) Determine the total mass of the oil. c) Within what radius is 75% of the mass? a) Mass = ( Area)( Density) ( πri ) r i + r i= πr b) Mass = dr 4,34 kg + r k πr c) dr (.75)(434) kg, k 78meters + r PROBLEM 4. Suppose the density of cars, in cars/mile, along a 3 mile stretch of the Mass Pike (during 3 rush hour) can be modeled by ρ( x) = (. x +. ), where x represents the number of miles from Boston. a) Write a function that gives the number of cars from Boston to a point x miles from Boston. b) Use this function to determine the total number of cars on this 3-mile stretch of road. a) C(x) = x ρ ( t) dt 3 3 b) C(3) = (.x +.) dt, 55cars i Page of 9
PROBLEM 5. Greater Boston can be approximated by a semi-circle of radius 8 miles from its center on the coast. Moving away from the center along a radius, the population density starts to decrease according to the data given in the table, where ρ( r ), measured in thousands of people/mile, is the population density at a distance r miles from the center. r (miles) 3 4 5 6 7 8 ρ( r ) 75 75 67.5 6 5.5 45 37.5 3.5 Using this data and Riemann sums, estimate the total population living in the 8 mile radius. 8 P(r) = Area x Density πr ρ( r ) r = 3,958 thousand people i= i i i or Area x Density πr ρ( r ) r 9 i= i i i = 4,54 thousand people. PROBLEM 6. A faucet is turned on at time t =, and t minutes later water is flowing into a barrel at a rate of R(t) = t + 4t gallons/minute, for t in [,5]. a) How much water was added to the barrel during these five minutes? b) How much water was added to the barrel during the third minute of the flow? c) Find the average rate of flow for the first five minutes. d) Find a one-minute interval in which the total flow was equal to the average flow. 5 dv 5 75 a) V(5) V() = dt ( t 4t) dt gallons dt = + = 3 b) V(3) V() = 49/3 gallons. c) Average rate of flow = 5 5 ( t + 4t) dt = 75 5 gallons minute d) solve x+ x ( 75 t + 4t) dt = We find x =., so the interval is [.,3.] 5 Page 3 of 9
PROBLEM 7. A mold of varying thickness is growing in the shape of a disk; the mold is thickest at the center of the disk and the thickness decreases as the distance from the center of the mold increases. At a distance R millimeters from the center, the thickness of the mold is 4-.3e R millimeters, for R. In the figure above, circles have been drawn of radii.5,.,.5 and. millimeters. The volume of the mold in the middle ring can be approximated in the following way: First we approximate the area of the ring by the product of the outer circumference and the thickness of the ring. Thus, Area of ring πr R = π ( 5. )( 5. ). Next we approximate the volume by using the height of the mold on the outer circumference. Note that this height is constant on such a circle. height = 4.3e.5. Finally, the volume of the mold in the middle ring is given by the height times the area, so Volume = (4.3e.5 ) π (.5)(.5). Using this technique with the other three subregions in the given figure, approximate the volume of the mold. Volume = (4.3e.5 ) π (.5)(.5) + (4.3e. ) π (.)(.5) + (4.3e.5 ) π (.5)(.5) + (4.3e. ) π (.)(.5) c) The answer to part a) is a Riemann sum with n = 4 over the interval [,] for what definite integral? ( 4.3e R )π R dr Page 4 of 9
d) Use this integral to find the volume of the solid. Volume = 34.45 cubic millimeters d) Use the same technique to set up a definite integral that would give the total number of people, in thousands, living in Diskville, if the population R miles from the center of Diskville is thousand people per square mile, and Diskville has a circular + R boundary 5 miles from the center of the town. 5 (π R) dr 88,79 people + R PROBLEM 8. The value V of a Tiffany lamp, worth $5 in 965, increases at the rate of 5% per year. Its value t years after 965 is given by V(t) = 5(.5) t. Find the average value of the lamp over the period 965. average value = 35 t 5(.5) dt 35 Page 5 of 9
PROBLEM 9. A bar of metal is cooling from o C to room temperature, which is o C. The temperature, H, of the bar t hours after it starts cooling is given by H(t) = + 98 e -.t. a) Find the temperature of the bar after one hour. b) Find the average value of the temperature of the bar over the first hour. c) Is your answer to part b) greater or smaller than the average of the temperatures at the beginning and end of the hour? Explain your answer in terms of the concavity of the graph of H(t). a) H(t) = + 98 e -.() = 96.74.t b) average value of H over [,] = ( + 98e ) dt 95.59 C c) H( ) + H( ) + 96. 74 = = 953. 37. The function is concave up. PROBLEM. After t hours, a population of bacteria is growing at the rate of t million bacteria per hour. Estimate the total increase in the bacteria population during the first hour. Change in population = t dt.4 millionbacteria. PROBLEM. The air density in kg/m 3 h meters above the surface of the earth is given by P = f(h). Find the mass of a cylindrical column of air meters in diameter and 5 km high. The mass of a horizontal air slice with thickness h ( Volume)( Density) = ( h) f ( h i ) π. n So the total mass π f ( hi ) h kg. i= definite integral 5 π f ( h) dh kg. As n approaches infinity, this sum represents the Page 6 of 9
PROBLEM. Water leaks out of a tank through a square hole with inch sides. At time t (in seconds), the velocity of water flowing through the hole is v = g(t) ft/sec. What is the definite integral that represents the total amount of water lost in the first minute? The area of the square hole is /44 square feet. The amount of water which has passed through in some time t can be thought of as a rectangular solid whose volume V = (Area)(Height) = (Area)(Velocity)(Time). Over t seconds, the amount of water lost ft ( g( t) ft / sec)( t sec) = g( t) t ft 44 44 3. Therefore, the total amount of water lost 44 g( t) dt ft 3, and as t goes to, this sum goes to the definite integral 6 44 g ( t) dt. PROBLEM 3. The rate of consumption of oil in the United States since 974 can be modeled by the function C(t) =.3 e.4t, where t is the number of years since 974 and C(t) is measured in billions of barrels of oil per year. Based on this model, how much oil was used in the United States in the decade beginning in 98 and ending 989? 5.4t Amount of oil = lim C t t ( ) =.3e dt 33.9 billion gallons of oil = t t 5 6 6 Page 7 of 9
PROBLEM 4. When a body moves a distance d along a straight line as the result of being acted upon by a force that has constant magnitude F in the direction of the motion, the work done by the force in moving the body is F times d: W = Fd. A leaky 5 pound bucket is lifted from the ground into the air by a worker pulling in feet of rope at a constant speed. The rope weighs.8 lb/ft. The bucket starts with gallons of water (6 lbs.) and leaks at a constant rate. It finishes draining just as it reaches the top. How much work is done in: a) lifting the water alone? b) lifting the water and the bucket? c) lifting the water, the bucket and the rope? a) When the bucket is x ft. off the ground, the water weighs x 4x F( x) = 6 = 6 lb. 5 4x ( 6 ft lb. 5 The work done is W = F x) dx = 6 dx = b) To lift the bucket and the water, the work is 6 + (5)()=6 ft. lb. c) The work in lifting all three: 4x F(x) = 6 + 5 + (.8)( x), so the total work is 76 ft. lb. 5 Page 8 of 9
PROBLEM 5. How much work does it take to pump the water from and upright and full right circular cylindrical tank of radius 5 ft. and height ft. to a level 4 ft. above the top of the tank? Consider moving "thin" slabs of water, with thickness y. The volume of a slab is about V = π (radius) (thickness) = π (5 ) ( y) = 5π y ft 3. The force required to lift this slab is its weight, F(y) = w V = 5πw( y) lb., where w is the weight of one cubic foot of water. The distance through which F(y) must act is about (4-y) feet, so the work done in lifting this slab 4 feet above the top of the tank is about W = 5 π w (4-y) ft. lb. So the total work is given by the definite integral Work = 5π w(4 y) dy =,5πw ft. lb. Page 9 of 9