Transcript Velocity and Speed: Working with Absolute Value

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
lim  v(ti )ti
t  0
b
Displacement =
 v(t )dt
a
=
s (b)  s ( a )
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).
Total Distance Traveled vs. Displacement
The velocity of a particle on the x-axis is modeled by the function,
x(t )  t. 3  4t
Find the Displacement and Total Distance Traveled of the particle on the interval, t  [ 0 , 6 ]
Beginning and Ending positions
b
 v (t ) dt
 s (b )  s ( a )
a
b
s(a) 
 v (t ) dt
 s (b )
a
b
s ( a )  s (b ) 
 v (t ) dt
a
Example:
Given:
ds
8
2
 v(t )  t 
dt
(t  1)2
on the interval for 0  t  5
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)
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 .
y
x
a) Find where the particle is at the end of the trip.
b) Find the total distance traveled by the particle.
General Strategy
1) Approximate what you want to find using a RIEMANN’S SUM
( rate * quantity )
ft
 sec
sec
dollars
 hours
hour
2) Write and solve the definite Integral
newtons
 meters
meter
Reading:
If
w(t ) is the rate of growth of a child in pounds per year
10
What does
 w(t )dt
represent.
5
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
 t 
C (t )  3.9 - 2.4 sin 

 12 
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 kilowatthours.
Population Density
The density function for the population in a certain city is
 ( x)  13.2 1  r  2 where r is the distance from the center of
the city in miles and ρ has units of thousands per square mile.
2
1
How many people live within a 20 mile radius of the city center.
y

Thickness Δr
Area = 2πr Δr



x









population =
people
 sq. miles 
sq. 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
2
flows at a distance r inches from the center is 8(10 - r ) 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.
in3
Flow in

min
Work
WORK = Force * distance
W=Fd
Hooke’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