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5046: Modeling with the Definite Integral

5046: Modeling with the Definite Integral. AP Calculus. Linear Motion Revisited. Displacement is the change in position from beginning point, a , to ending point, b . Incremental change = rate of change * increment of time v ( t ) *  t Displacement = .

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5046: Modeling with the Definite Integral

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  1. 5046: Modeling with the Definite Integral AP Calculus

  2. Linear Motion Revisited Displacement is the change in position from beginning point, a , to ending point, b . Incremental change = rate of change * increment of time v ( t ) *  t Displacement =

  3. Velocity and Speed: Working with Absolute Value The Definite Integral of velocity is NET distance (DISPLACEMENT). DEFN: Speed is the Absolute Value of Velocity. The Definite Integral of Speed is TOTAL distance. (ODOMETER).

  4. Total Distance Traveled vs. Displacement The velocity of a particle on the x-axis is modeled by the function, . Find the Displacement and Total Distance Traveled of the particle on the interval, t  [ 0 , 6 ]

  5. Beginning and Ending positions

  6. Example: Given: Write the integral that represents the displacement of the particle given the following information. If s(0) = 9 , find s(5) If s(5) = 81 , find s(0)

  7. p. 386 # 20 The graph of the velocity of a particle moving along the x-axis is given. The particle starts at x = 2 when t = 0 . Find where the particle is at the end of the trip. Find the total distance traveled by the particle.

  8. General Strategy 1) Approximate what you want to find using a RIEMANN’S SUM ( rate * quantity ) 2) Write and solve the definite Integral

  9. Reading: If is the rate of growth of a child in pounds per year What does represent. If water leaks from a tank at a rate of r (t) gallons per minute at time t , write a definite integral to find the totalamount of water that leaks out in between the hours 2 and 5 . A honey bee population starts with 100 bees and increases at a rate of n /(t) bees per week. Write a definite integral to give the population after 15 weeks.

  10. # 22/p.386 The rate at which your home consumes electricity is measured in kilowatts. If your home consumes electricity at a 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 average daily consumption for this home, measured in kilowatt-hours.

  11. Population Density The density function for the population in a certain city is where r is the distance from the center of the city in miles and ρ has units of thousands per square mile. How many people live within a 20 mile radius of the city center. Thickness Δr Area = 2πr Δr

  12. #24/p.387 Oil flows through a cylindrical pipe of radius 3 in., but friction from the pipe slows the flow toward the outer edge. The speed at which the oil flows at a distance r inches from the center is inches per second. In a plane cross section of the pipe, a thin ring with thickness Δr at a distance r inches from the center approximates a rectangle when it is straightened out. Find the area for the ring. Set up an evaluate a Definite Integral that will give the rate at which the oil is flowing through the pipe.

  13. Work WORK = Force * distance W = F d Hooke’s Law: If F(x) represents the force in Newtonsrequired to stretch a spring x meters from its natural length. Then F(x)=kx If it takes a force of 10 N to stretch a spring 2 m beyond its natural length. How much WORK is done in stretching the spring 4 m from its natural length.

  14. Last Update : 03/19/2012 Assignment: p. 386 # 1 -11 odd , 12-16 , 17 – 23 odd , 29

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