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 =

0lim ( )i it

v t t

( ) = ( ) ( )b

a

v t dt s b s a

Velocity and Speed: Working with Absolute ValueThe 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).

Total Distance Traveled vs. DisplacementThe 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 ]

3( ) 4x t t t

Beginning and Ending positions

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

b

a

b

a

b

a

v t dt s b s a

s a v t dt s b

s a s b v t dt

22

8( ) on the interval for 0 5 ( 1)

ds v t t tdt t

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)

x

y

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 .

a) Find where the particle is at the end of the trip.

b) Find the total distance traveled by the particle.

General Strategy1) Approximate what you want to find using a RIEMANN’S SUM

( rate * quantity )

2) Write and solve the definite Integral

sec secft dollars newtonshours meters

hour meter

Reading: If is the rate of growth of a child in pounds per year

What does represent.

( )w t10

5

( )w t dt

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 total amount 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.

# 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.

( ) 3.9 - 2.4sin 12

tC t

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.

1

2 2( ) 13.2 1x r

Thickness ΔrArea = 2πr Δr

x

y

population = . .

people sq milessq mile

#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.

28(10 - )r

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. 3

Flow in minin

Work

WORK = Force * distance W = F dHooke’s Law: If F(x) represents the force in Newtons required 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.

Last Update :

03/19/2012

Assignment:

p. 386 # 1 -11 odd , 12-16 , 17 – 23 odd , 29