Areas Between Curves and Other Applications of Integration

Intro to CalculusName:

Date: April 8, 2016

Quiz Review

Areas between curves and other applications of integration

Example:

In the Annual Energy Outlook, the U.S. Energy Information Administration projected the rate of consumption C (in quadrillions of Btu per year) of petroleum to model:

Where t = 0 represents January 1, 2000. If the actual rate of consumption more closely followed the model:

How much petroleum would be saved?

Good viewing window: ,

1. Sketch the region bounded by the graphs of the functions and find the area of the region analytically.

2. Sketch the region bounded by the graphs of the functions and find the area of the region analytically.

3. The marginal cost for producing x units of a product is modeled by .

It costs $60 to produce one unit. Find the cost of producing 100 units.

4. The rate at which your home consumes electricity is measured in kilowatts. If your home consumes electricity at the rate of 1 kilowatt for 1 hour, you will be charged for 1 “kilowatt-hour” of electricity. Suppose that the average consumption rate for a certain home is modeled by the function , where C(t) is measured in kilowatts and t is the number of hours past midnight. Find the daily consumption for this home, measured in kilowatt-hours. Show work involving integrals.

5. If a force of 80 N is required to hold a spring 0.3 m beyond its natural length, how much work does it take to stretch the spring this far? How much work does it take to stretch the spring an additional meter? Show work involving integrals.

6. Find the area of the region bounded by the following curves analytically.

Hint: Use symmetry to make your calculations more efficient.

7. The graph of the velocity of a particle moving on the x-axis is given below. The particle starts at x = 2 when t = 0.

a. Find where the particle is at the end of its trip. Show work involving integrals.

b. Find the total distance traveled by the particle. Show work involving integrals.

8. A toy car slides down a ramp and coasts to a stop after 5 seconds. Its velocity from t = 0 to t = 5 is modeled by ft/sec. How far does it travel?

9. Find the area of the shaded regions below by setting up and evaluating integrals analytically.

a. b.

c. d.

10. Find the average value of the functions below on the given interval.

a. [0, 9]

b. [1, 3]

11. Rich Wholefood Sales, a manufacturer of cookies, stores its cases of cookies in an air-conditioned warehouse for shipment every 14 days. Rich tries to keep 600 cases on reserve to meet occasional peaks in demand, so a typical 14-day inventory function is , . The holding cost for each case is 4 cents per day. Find Rich’s average daily inventory and average daily holding cost.

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Answers

1. 2. 3. $3,919.02

4. 93.6 kilowatt-hours5. 12 Nm (J) and 213.3 Nm (J)

6. 87. a. x = 2b. 4 m8. ft

9. a. b. c. 16d.

10. a. 79b. 1090

11. a. 4800 casesb. $192 per day

From pg. 395 #1, 5, 13, 14, 18 and pg. 386 #18, 22 and pg. 430 #1, 34