Using Integrals
Rate of consumption of oil in the United States during the 1980s (in billions of barrels per year) is modeled by the function R(t)=27.08e^(t/25) where t is the number of years after January 1, 1980. Find the total consumption of oil in the United States during the 1980s.
332.96 billion barrels
Construction workers are pouring concrete at a rate modeled by C(t) measured in cubic feet per minute and t is measured in minutes since the start of the workday. When the day begins, there was already 60 cubic feet of concrete that has been poured from the day before. Write, but do not solve, an equation involving an integral to find the time x when the amount of concrete poured has reached a total of 100 cubic feet.
60 + integral 0 to x C(t) dt = 100
The atmospheric pressure of the air changes with height above sea level. The rate of change of the air pressure at a given height above sea level can be measured by the differentiable function f(h), in psi per meter, where h is measured in meters. What are the units of the integral of f(h)dh?
psi
Find the average value of the function f(x) = -x^2-2x+5 [-4,0]
11/3 = 3.66
The temperature of water in a bathtub at time t is modeled by a strictly decreasing, twice-differentiable function W, where W(t) is measured in degrees Fahrenheit and t is measured in minutes. The water is cooling for 30 minutes, beginning at time t = 0. Values of W(t) at selected times t are given in the table below. What would the integral from 0 to 30 of W'(t) dt means in the context of the problem.
t minutes 0 10 20 30
W(t) degrees 100 89 81 75
Water cooled 25 degrees in the first 30 minutes
The ocean depth near the shore is changing at a rate modeled by R(t) = 3.0368 sin (t*pi/6), measured in feet per hour t hours after 10 A.M. If the depth is 10 feet at 10 A.M., how deep is the water at 1:00 p.m .?
15.79 ft
Rain is falling at a rate modeled by R(t) measured in inches per hour and t is measured in hours since noon. By noon, there has been 2 inches of rain that has already fallen that day. Write, but do not solve, an equation involving an integral to find the time A when the amount of rain that has fallen for the day has reached a total of 3 inches.
2+Intergal from 0 to A R(t) dt = 3
The atmospheric pressure of the air changes with height above sea level. The rate of change of the height above sea level for a given air pressure can be measured by the differentiable function f(p), in feet per psi, where p is measured in psi. What are the units of f'(p)?
feet per psi per psi
Find the average value of the function -x^4+2x^2+4 on the interval [-2,1]
19/5 = 3.8
The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of time t. A table of selected values of R(t), for the time interval 0 ≤ t ≤ 90 minutes, is shown below.
T(t) minutes 0 20 30 50 60 90
R(t) gallons per minute 15 60 45 60 70 75
Using correct units, interpret the meaning of the the intergal from 0 to 30 R(t) dt in context of the problem.
TOTAL gallons used for the first 30 minutes of flight
The temperature outside during a 12-hour period is given by the function T(h)= 60- 5 cos (pi*h/8) for 0<h<12. Where T(h) is measured in degrees Fahrenheit and h is measured in hours. Find the average temperature, to the nearest degree Fahrenheit, between h = 2 and h =9.
0 ≤h ≤ 12
62 degrees Farenheit
The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R. A table of selected values of R(t), for the time interval 0 ≤ t ≤ 90 minutes, is shown below.
t minutes 0 20 30 50 60 90
R(t) gallons per minute 15 60 45 60 70 75
Use a left-point Riemann sum, with five subintervals, to approximate the integral from 0 to 90 R(t) dt.
4500 gallons
The atmospheric pressure of the air changes with height above sea level. The rate of change of the height above sea level for a given air pressure can be measured by the differentiable function f(p), in feet per psi, where p is measured in psi. What are the units of the integral for f(p)dp?
feet
The temperature F, in degrees Fahrenheit (°F), of a cake coming out of an oven is given by 74 + 92e^-0.062t. To the nearest degree, what is the average temperature of the cake between t = 0 and t = 8 minutes?
147 degrees
A parking garage has 230 cars in it when it opens at 8 AM (t=0). On the interval 0≤t≤10, cars enter the parking garage at a rate modeled by the function E(t)=58cos(0.163t-0.642) and leave the parking garage at a rate modeled by the function L(t)=65cos(0.28lt)+7.1 beginning at 9 AM and continuing until 6 PM (t=10). Both E(t) and L(t) are measured in cars per hour while t is measured in hours. If E'(5) = -1.627 interpret the meaning of this in context (be sure to include correct units)
At hour 5, the rate at which cars are entering the garage is decreasing at a rate of 1.627 cars per hour per hour
The rate of gallons of gasoline used per kilometer by a car to travel x kilometers is modeled by the function g(x) = .15-.15e^(-x/2)-.075xe^(-x/2). If the car started with 36 gallons, how many gallons are left after driving 200 kilometers?
6.45 gallons
On a late winter day in Avon, Indiana, the temperature in °F, t hours after 9 A.M. can be modeled by the function function T(t)=55+14cos (t*pi/12). Find the average temperature during the period from 9 A.M. to 9 P.M.
55 degrees
The differentiable function f measures the rate at which the temperature of a turkey in an oven is changing, in degrees Celsius per minute, where t is measured in minutes. What are the units of
1/5 * integral 2 to 7 of f'(t) dt.
degrees Celcius per minute per minute
The function f is defined by f(x) = (x^2 - 4x) sin (x -1). What is the average rate of change of f on the interval [4, 8]? You may use a calculator and round your answer to the nearest thousandth.
5.256