CALCULUS BC
CALCULUS BC
WORKSHEET ON APPLICATIONS OF THE DEFINITE INTEGRAL
Work the following on notebook paper. Use your calculator, and give decimal answers correct to three decimal places.
1. Two runners, A and B, run on a straight racetrack for
[pic] seconds. The graph shown, which consists
of two line segments, shows the velocity, in meters
per second, of Runner A. The velocity, in meters
per second, of Runner B is given by the function
defined by [pic].
(a) Find the total distance run by Runner A over the
time interval [pic] seconds.
(b) Find the total distance run by Runner B over the
time interval [pic] seconds.
2. The rate at which water flows out of a pipe, in gallons per hour, is
given by a differentiable function R of time t. The table shows
the rate as measured every 3 hours for a 24-hour period.
(a) Use a midpoint Riemann sum with 4 subdivisions of equal
length to approximate [pic]. Using correct units,
explain the meaning of your answer in terms of water flow.
(b) The rate of water flow [pic] can be approximated by
[pic] Use [pic] to approximate
the average rate of water flow during the 24-hour time
period. Indicate units of measure.
3. A bar of metal is cooling from 1000° C to room temperature, 20° C. The temperature, H, of the
bar t minutes after it starts cooling is given, in ° C, by [pic]
(a) Find the temperature of the bar at the end of one hour.
(b) Find the average value of the temperature over the first hour.
4. A cup of coffee at 90° C is put into a 20° C room when t = 0. The coffee’s temperature is
changing at a rate of [pic]° C per minute, with t in minutes.
(a) Find the coffee’s temperature when t = 10.
(b) Find the average rate of change of the temperature of the coffee over the first ten minutes.
5. Methane is produced in a cave at the rate of [pic] liters per hour at time t hours.
The initial amount of methane in the cave at time t = 0 is 20 liters. At time t = 8 hours, a
pump begins to remove the methane at a constant rate of 1.5 liters per hour.
(a) At what time t during the time interval [pic] hours is the amount of methane
increasing most rapidly?
(b) What is the total amount of methane in the cave at time t = 8 hours?
(c) What is the average rate of methane accumulation in the cave over the time
interval [pic] hours?
(d) What is the absolute maximum amount of methane in the cave over the time
interval [pic] hour.
TURN->>>
6. Water flows in and out of a storage tank. A graph of the rate of
change [pic] of the volume of water in the tank, in liters per day,
is shown. If the amount of water in the tank at time t = 0 is 25,000L,
use the Midpoint Rule with four equal subintervals to estimate the
amount of water four days later.
7. Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks
out of the tank at the rate of [pic] gallons per minute, for [pic] minutes. At time t = 0, the
tank contains 30 gallons of water.
(a) How many gallons of water leak out of the tank from time t = 0 to t = 3 minutes?
(b) How many gallons of water are in the tank at time t = 3 minutes?
(c) Write an expression for [pic], the total number of gallons of water in the tank at time t.
(d) At what time t, for [pic], is the amount of water in the tank a maximum? Justify your
answer.
8. The number of gallons, [pic], of a pollutant in a lake changes at the rate [pic]
gallons per day, where t is measured in days. There are 50 gallons of the pollutant in the lake
at time t = 0. The lake is considered to be safe when it contains 40 gallons or less of pollutant.
(a) Is the amount of pollutant increasing at time t = 9? Why or why not?
(b) For what value of t will the number of gallons of pollutant be at its minimum? Justify your
answer.
(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your
answer.
(d) An investigator uses the tangent line approximation to [pic] at t = 0 as a model for the
amount of pollutant in the lake. At what time t does this model predict that the lake
becomes safe?
9. A particle moves along the x-axis so that its velocity v at any time t, for [pic], is given
by [pic]. At time t = 0, the particle is at the point [pic].
(a) For what values of t, [pic], is the particle moving to the right? to the left?
(b) Find the position of the particle at t = 4.
(c) Find the total distance traveled by the particle from t = 0 to t = 4.
Answers
1. (a) 85 meters (b) 83.336 meters
2. (a) 258.6 gallons (b) 10.785 gal/hr
3. (a) 22.429°C (b) 182.928°C
4. (a) 45.752°C (b) – 4.425°C/min
5. (a) 2 hours (b) 30.129 liters (c) 0.266 liters/hr (d) 32.174 liters
6. 28.250 liters
7. (a) [pic] gallons (b) [pic] gallons (c) [pic]
(d) 63 minutes
8. (a) decreasing, not increasing (b) 30.174 days (c) The lake is safe.
(d) 5 days
9. (a) Right for [pic] and left for [pic]
(b) 8.671 (c) 10.542
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