Mathl15

Notes

Integrals

The

integral

of a function,

(x),

is

symbolized

by

I

tOV*.

The

differential

dx indicates the

variable

of

integral

is

always

written

to the

far right. The expression

between the integral

sign and

the

differential

is the

expression

to be integrated.

This

representation

is for a function of

a single

variable,

usually

studied

in

Calc

I & U.

In multivariable

calculus

(calc

III), the function

to be

integrated is usually dependent

on several

variables,

i.e.,

(x,y)

or

(x,y,z)

*d multiple

integrals appear. For

exampte,

[lf{*,y)dxdy

and

l![f

t*,t,r)tudydz

represent

a double and

triple

integral in calculus III and can be

done by two and

three separate

integrations,

respectively.

They are

done by separate

integrations

starting

from the

inside out.

For example,

in the case of the

triple

integral,

the

frst integration is done

with respect

to x,

then

with

respect

to

y,

and

finally

with respect to z. The other variables

in the expression

f(x,y,z) are

treated

as constants

when

performing

the integration with respect to one of

them. For

multiple

integrals,

the

order of

integrations is always done from the

inner

most

variable out to the

variable

on the

extreme

right.

(These

variables

are indicated

by

their differentials).

These are examples

of

Indelinite

Integrals, since no limits appear on the integration symbol. So,

the results

will be

variable expressions with

unknown constants

(assuming

no conditions

on the

problem

are

given).

Definite Integrals have

limits

on

the integration symbol

(lower

& higher).

See

my handout

on the definite

integral and

what it

represents.

Since

integration

is

the

reverse

process

of finding a derivative,

it is

called anti-differentiation

by

most texts &

teachers, as

wel[.

So,

going

from a

derivative to its

parent

function

will involve integrating the derivative.

However,

for indefinite

integrals, the result

will

contain many answers

since

an

unknown

constant

will be

present.

In symbols,

!

f

ti*:F(x)

+

C. I

use

F(x) in the result since that

function will differ

from f(x). The constant C

is arbitrary and could take on an infinite number of

values, if no other

information is

given

in the

problem.

It will always be the case that

F'(x):

(x).

This

is

a fundamental relationship that ties both

processes

together. This

is

one

of the fundamental theorems of calculus. For example,

tz**-x'

+

C.

Note

that

the derivative of x2

is 2x,

which

is the expression in the integral.

However,

only

the simplest of integrals

can be done by

guess

work.

Rules

are established for

integration

so

more complicated integrals can

be evaluated. There are expressions

that are very

complicated &

special techniques are

studied

(calc

II) and others that can't

be

done.

However,

most applications deal

with definite integrals, theirvalues can be approximated

very closely. The

calculator will do

this

for us

very

quickly

(see

handout on the definite integral).