Calculus 7.1

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7.1 Integral as Net Change
Greg Kelly, Hanford High School, Richland,
Washington
Photo by Vickie Kelly, 2006
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A honey bee makes several trips from the hive to
a flower garden.
The velocity graph is shown below.
What is the total distance traveled by the bee?
700 feet
200ft
200ft
200ft
100ft
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What is the displacement of the bee?
100 feet towards the hive
200ft
200ft
-200ft
-100ft
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To find the displacement (position shift) from
the velocity function, we just integrate the
function. The negative areas below the x-axis
subtract from the total displacement.
To find distance traveled we have to use absolute
value.
Find the roots of the velocity equation and
integrate in pieces, just like when we found the
area between a curve and the x-axis. (Take the
absolute value of each integral.)
Or you can use your calculator to integrate the
absolute value of the velocity function.
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Displacement
Distance Traveled
velocity graph
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Linear Motion

• V(t) is the velocity in m/sec of a particle
  moving along the x-axis and starting at the
  position, s(0) 8.
• Determine when the particle is moving to the
  right, to the left, and stopped.
• Find the particles displacement for the given
  time interval and its final position.
• Find the total distance traveled by the particle.

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Linear Motion

• V(t) is the velocity in m/sec of a particle
  moving along the x-axis and starting at the
  position, s(0) 8.
• Determine when the particle is moving to the
  right, to the left, and stopped.
• Find the particles displacement for the given
  time interval and its final position.
• c) Find the total distance traveled by the
  particle.

Particle is moving left on 1 lt t lt 2, stopped at
t 2 and moving right on 2 lt t lt 4.
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Effects of Acceleration

• A car moving with initial velocity of 5 mph
  accelerates at the rate of a(t) 2.4 t mph per
  second for 8 seconds.
• How fast is the car going when the 8 seconds are
  up?
• How far did the car travel during those 8 seconds?

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Effects of Acceleration

• A car moving with initial velocity of 5 mph
  accelerates at the rate of a(t) 2.4 t mph per
  second for 8 seconds.
• How fast is the car going when the 8 seconds are
  up?
• b) How far did the car travel during those 8
  seconds?

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In the linear motion equation
V(t) is a function of time.
For a very small change in time, V(t) can be
considered a constant.
We add up all the small changes in S to get the
total distance.
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We add up all the small changes in S to get the
total distance.
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This same technique is used in many different
real-life problems.
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Example 5
National Potato Consumption
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Example 5
National Potato Consumption
We add up all these small amounts to get the
total consumption
From the beginning of 1972 to the end of 1973
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Work
Calculating the work is easy when the force and
distance are constant.
When the amount of force varies, we get to use
calculus!
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Hookes law for springs
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Hookes law for springs
Example 7
It takes 10 Newtons to stretch a spring 2 meters
beyond its natural length.
How much work is done stretching the spring to 4
meters beyond its natural length?
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How much work is done stretching the spring to 4
meters beyond its natural length?
For a very small change in x, the force is
constant.
p
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A Bit of Work
It takes a force of 16 N to stretch a spring 4 m
beyond its natural length. How much work is done
in stretching the spring 9 m from its natural
length?
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A Bit of Work
It takes a force of 16 N to stretch a spring 4 m
beyond its natural length. How much work is done
in stretching the spring 9 m from its natural
length?