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7.1 Integral as Net Change

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7.1 Integral as Net Change. 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. What is the displacement of the bee?. 100 feet towards the hive.

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slide3

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

slide4

What is the displacement of the bee?

100 feet towards the hive

200ft

200ft

-200ft

-100ft

slide5

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.

slide6

Every AP exam I have seen has had at least one problem requiring students to interpret velocity and position graphs.

position graph

Displacement:

Distance Traveled:

velocity graph

linear motion
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 particle’s displacement for the given time interval and its final position.
  • Find the total distance traveled by the particle.

Particle is moving left on 1 < t < 2, stopped at t = 2 and moving right on 2 < t < 4.

effects of acceleration
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?
slide9

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.

slide10

As the number of subintervals becomes infinitely large (and the width becomes infinitely small), we have integration.

We add up all the small changes in S to get the total distance.

slide12

The rate of potato consumption for a particular country was:

where t is the number of years since 1970 and C is in millions of bushels per year.

For a small , the rate of consumption is constant.

The amount consumed during that short time is .

Example 5:

National Potato Consumption

slide13

The amount consumed during that short time is .

million bushels

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:

slide14

Work:

Calculating the work is easy when the force and distance are constant.

When the amount of force varies, we get to use calculus!

slide15

x= distance that the spring is extended beyond its natural length

k= spring constant

Hooke’s law for springs:

slide16

F=10 N

x=2 M

Hooke’s 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?

slide17

F(x)

x=4 M

newton-meters

joules

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

a bit of work
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?