AreasBetweencarvemotivatingExamplei

AreasBetweencarvemotivatingExamplei

Suppose there are two populations A and B

Pact

PB t

Size of population A at time t Size of population B at time t

Assume Pa t

t and P's t

sin t

Growth

rate at

populations

and

PO

PB o

Interpret

y Pt

E

ZIT

the array between the

and y Pig t between

curves

C o and

4 Calculate this area

yet Pa Ct

?

3

Area between

curves

Area

I

Area

I

0

it

y

si ult

PB t

Area A Area B

Area

Area

Pict dt

Pp t dt

Area Xi

zit

Jp e dt

IT

Area

ZIT

J P'Bit It

IT

it

f a'HI PBHI d

0

under X axis so area is

counted negatively

zit

fLpa t PBHI dt

IT

Area between

Curves

FTC

conclusion

zit

JP t

o

P's t

P It

on

dt

21T

PB t

0,21T

Pact

P Ct

OL

Palo Polo

Pa 21T I PB 121T

Interpretation

Area between arrives between

C o and t zit

Difference between sizes of populations

A and B at time C zit

21T

f t sin t dt

O

21T

t't cos t to

I att T

211 2

cos zit

oc so

Q

Given 7 and g

functions on

ab

what is

the total

area enclosedy by y

tea

and

y gcse

between t

a and t b

Picture

i y g ca

i

Land

a

b

y 1x

Important observation

Total

Fcx

area enclosed

Area A

Area B Area c Area D

gcse dx

Area A

Area B Area c Area D

Strategy

Remark

Find area of each region individually

Not

very

useful in

practice

Total area enclosed

y y text and gcx between

C a and C b

Total area enclosed

by y Flat g

and x axis between

E a and t b

flux guild

a

Calculate by doing sign analysis on 7cal gca

and calculating

dat Jatin ga

onsubeiffheevar

Example

Find total area enclosed by y

between C O and E Iz

sin x

I Do sign analysis on Siu x cos x

and y csscx

It

sincal

cos x

o

Ys

I

tank I

x

3

sincx cos x

Cts ou Co

3

IT

4

sin x cos x

I

2

Siulo Crisco co

Siu Iz

sci o

There are

area For

Picture

two regions we must calculate

y

to

y

4 Iz

siu x sea

Area A

14

J Sinixt

T

casca

dx

in 174 Casca EO

on Co

3

IT

C costa Sin la

o

Area B

cos I sinc

2

I

1172

J Sinixt

VE I

cos x dx

cos 107 sin Col

in x Cns x

Ct

4

O

C costa Sin La f onset sin Il

IT I

The

4

cos

I since

Total area enclosed

2 VE l

m

To calculate

total area enclosed

by y fCx and y g x between

E a and t b

I Do sign analysis on 7cal g Coo ou carb

7 Calculate 11,7cm ga da f on each sub interval

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