Lecture 4.3 Integral as Net Change

8/7/2019 Lecture 4.3 Integral as Net Change

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ft

min

minutes

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?

200ft

200ft

200ft

100ft

200 200 200 100 700 ! 700 feet

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ft

min

minutes

What is the displacement of the bee?

200ft

-200ft

200ft

-100ft

200 200 200 100 100 !

100 feet towards the hive

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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 totaldisplacement.

Displacementb

a

V t dt !

Distance Traveledb

a

t dt! 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 curveand 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. p


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Theimage cannotbed isplayed.Your computer may nothaveenough memory toopen theimage,or theimage may havebeen corrupted.Restart your computer,and then open thefile again.If thered x stillappears,you may havetodeletethe imageand then insertit again.Theimage cannotbed isplayed.Your computer may nothaveenough memory toopen theimage,or theimage may havebeen corrupted.Restart your computer,and then open thefile again.If thered x stillappears,you may havetodeletethe imageand then insertit again.Theimage cannotbed isplayed.Your computer may nothaveenough memory toopen theimage,or theimage may havebeen corrupted.Restart your computer,and then open thefile again.If thered x stillappears,you may havetodeletethe imageand then insertit again.Theimage cannotbed isplayed.Your computer may nothaveenough memory toopen theimage,or theimage may havebeen corrupted.Restart your computer,and then open thefile again.If thered x stillappears,you may havetodeletethe imageand then insertit again.

Theimage cannotbed isplayed.Your computer may nothaveenough memory toopen theimage,or theimage may havebeen corrupted.Restart your computer,and then open thefile again.If thered x stillappears,you may havetodeletethe imageand then insertit again.Theimage cannotbed isplayed.Your computer may nothaveenough memory toopen theimage,or theimage may havebeen corrupted.Restart your computer,and then open thefile again.If thered x stillappears,you may havetodeletethe imageand then insertit again.Theimage cannotbed isplayed.Your computer may nothaveenough memory toopen theimage,or theimage may havebeen corrupted.Restart your computer,and then open thefile again.If thered x stillappears,you may havetodeletethe imageand then insertit again.

velocity graphTheimage cannotbed isplayed.Your computer may nothaveenough memory toopen theimage,or theimage may havebeen corrupted.Restart your computer,and then open thefile again.If thered x stillappears,you may havetodeletethe imageand then insertit again.

position graph

1

2

1

2

1

2

Displacement:

1 11 2 1

2 2 !

Distance Traveled:

1 11 2 4

2 2 !

Every AP exam I have seenhas had at least one

problem requiring students

to interpret velocity and

position graphs.

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In the linear motion equation:

dS V tdt

! V(t) is a function of time.

For a very small change in time, V(t) can be

considered a constant. dS V t dt!

S V t t ( ! ( We add up all the small changes in S to getthe total distance.

1 2 3S V t V t V t ! ( ( (

1 2 3S V V V t! (

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S V t t( ! ( We add up all the small changes in S to getthe total distance.

1 2 3S V t V t V t ! ( ( (

1 2 3

S V V V t! (

1

k

n

n

S V t

!

! (

1

n

n

S V t

g

!

! (

S V t dt!

As the number of subintervals becomes

infinitely large (and the width becomes

infinitely small), we have integration.

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This same technique is used in many different real-life

problems.

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Example 5: National Potato Consumption

The rate of potato consumption

for a particular country was:

2.2 1.1tC t !

where t is the number of years

since 1970 and is in millions

of bushels per year.

For a small , the rate of consumption is constant.t(

The amount consumed during that short time is . C t t (

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Example 5: National Potato Consumption

2.2 1.1

tC t !

The amount consumed during that short time is . C t t (

We add up all these small

amounts to get the totalconsumption:

total consumption C t dt!

4

2

2.2 1.1tdt

4

2

12.2 1.1

ln1.1

tt!

From the beginning of 1972 to

the end of 1973:

7.066}million

bushels

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Hookes law for springs: F kx!

x = distance that

the spring is

extended beyond

its natural length

k= spring

constant

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Hookes law for springs: F kx!

Example 7:

It takes 10 Newtons to stretch a

spring 2 meters beyond its natural

length.

F=10 N

x=2 M

10 2k!

5 k! 5F x!

How much work is done stretching

the spring to 4 meters beyond its

natural length?

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F(x)

x=4 M

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.

dw F x dx!

5 dw x dx!

5 dw x dx!

4

0

5 W x dx!

4

2

0

5

2W x!

40W ! newton-meters

40W ! joules

5F x x!

T

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Lecture 4.3 Integral as Net Change