Mathl

15

Notes

The

Definite

Integral

Example:

Since

many

of

the

popular applications

of integration

to

business

involve

the definite

integral,

we

need

know

how

to

get

this

number.

The

definite

integral,

!

tt*l*,

represents

the

numerical

value

of

the area enclosed

by

the

curve

y:(x) and

the

x-axis.

ln

business,

fC)

rf *$ty

involve

rates

of

aprofit,

cost,

revenue, or a

u"ppf,la"**a

fi]nction.

The results

of these

integrals

witl

grve

the

value-of

these

functions

over

the limits of

ini"l#tio".

This

number

is

positive, if

(x)

lies entirelyabove

the

x-axis

between

a and b.

Negative,

if more area

i,

fio*

the

x-uris

than

abwe

between

a and

b.

Zero,

if

equal

amounts

of

area are

above and

below

between x=a

andx:b

(i.e.,5sin(x)

from

a1'o2n).

3

Method

1:

lntegration

&

direct

evaluation.

Let's

take,

J{r'

*

2x2

-

57&.

I

Integrating

&

evaluating:

1x4

t4

+el3)x3

-5x)

I

l:1ttt+

+5413-15)

<ll4+213

-5):32811b27.33

(two

decimal

places).

Note:

Not

all

definite

integrals

can

be done

this

way.

Some

functions

are

too

complicated

& can

not be

integrated,

so,

in those

cases,

the

cilculator

method

must

be

used

to approximate

them.

You

will

need to

know

this method-

iheoretically,

all

definite

integrals

can be

done

by

the

calculator

method,

however,

the

answer may

not be exact.

For

exampti

in

tne

above

.*r*pl",

the

exact

answer

is328ll2.

The calculator

will

give

the

decimal form.

So,

if

an

exact

answer

is

required,

yormust

grve328ll2 and

not

round

offto

a certain

number

of decimal

places.

Method

2:

Using

your

calculator.

Code

in

the

]F=

menu

the

function

to be

integrated.

This is

located

between

the

integpl

symbol

and

the

dx.

This is

called

the

integrand.

In the

exarnple

above,

that

will

be x3

+2x2

'5.

You need

to b-

very

careful

coding

in

more complicated

expressions.

(see

note below)

Make

sure

that

your

x

window

spans

the

range

of

the limits

of

integration

(i.e.,

in this

exanrple,

the

x window

mnge

must

include

I

to

3).

This

is u

"o*-oi

mistake

made

by

many students.

Any range

including

these

o.rrib".r

will do,

however,

I try

to use

a

small

range.

This

will

glve

me a

better

graph,

if

desired.

Go to

2nd

trace

(calc)

to

menu

7.

Insert

the

lower

limit of

1, enter,

then

the upper

limit

of

3, enter.

ffts

answer

will be

displayed

at the

bottom

of

the

display.

Note:

If

you

would

like

to

have

a sketch

&

visualize

the

area enclosed

by

the

curve &

the x-a:ris,

go

to

2d.graph

to

your

table.

Insert

I

&3

to

get

their

y-values and

then adjust

the

y

window

according.

A sketch

is

very nice

to

sei

since

the numerical

value of

the

definite

integral

is displayed

by

a shaded

area.

This

may

enhance

your

understanding

of

one

meaning

for the

definite

integral.

Note:

tn

a

regular calculus

course,

definite

integrals

could

represent

many

other

quantities such

as

volume,

length,

work,

pressure,

distance,

just

to

name

a

few.

b

Note:

At

the bottom

of

thedisplay,

you

should

see

!

tt*l*

=27.33,the

same

answer

as before.

Note:

The

calculator

has a

distinct

advantage

,irr""

y*

do

not

need know

any

integration

rules. You

may

use this

method

on

all

problems that arise.

However,

in a Calculus

I, teachers

usually

require

you

to

know all

integration

rules

&

techniques.

Note:

Coding

in the

expression

seems

to be

the

biggest

problem for students.

Misuse of

symbols

of

inclusion,

like

parentheses

are co**on

mistakes

along

with

not including the x

limits

of

integration

in the

x

window.