Class

Meeting #21 Integration

(

antiderivatives)

-This

is

the

reverse

process

of

getting

derivatives.

-Going

from

a

derivative

to the original function(s).

-Consider

the

equation,

dyldx

=

2x

(differential

equation). Guess

the

y.

-Most

general,

y:

xz

*

C, where C

is

any

constant.

-We

say that x2

+

C is

the

integral

or antiderivative of 2x with respect

to

x.

f

-In

symbols,

J

2*

d*

=

xt

+

C, Note the location

of the

derivative of

x'+

C.

-Therefore,

we have

an

important

tie between the

two

major

operations

of

Calculus,

namely,

dif[erentiation

&

integration.

-This

is known

as the Fundamental Theorem

of

Calculus

(2od)

'

n

-[n

general,

d/dx

I

f1x;ax

=

(x).

t)

-The

first

Fundamental

Theorem ties the

process

of integration with the

area under a

curve.

-When

integrating

the

integration

symbol

is

always

on the

far left

&

the

differential

(dx)

symbol

is

always on the far right.

-The

differential indicates

the variable

of

integration

& is meaningless

without it.

*The

derivative is located

between the

integration

symbol

(far

left)

& the differential

(far

right).

So,

you

always have

a way of checking if

you

integrated

properly.

-Guessing

y

for

dy/dx

=

*'''

y"'

*uy not

be so easy,

so, we need

to

have rules

that

eliminate

the

guess

work...see

handout

sheet & cover examples...

-Before

you

use

the

rules

of integration

(on

handout),

the integral must

be

set-up

properly.

(will

show)

-Let's

cover some

of the

rules &

related

examples.