Class
Meeting #2J
The Dglinile
l$tpsral
-We
will tieArea with
the
integration.process.
-The
definite
integral
will
give
the area betwsen
the curve &
the
x-axis.
(may
be
positive
or
negative).
-Let
ffix)
be
a continuous
function over
a closed interval
[a;b].
Partition
the
interval
[a;b]
into
n
subintervals,
each
of
lengt6
4;s:(b-aln.
Choose
any >rC in
each
subinterval
and
form
t}re
R.iemann
Suur"
l'v
f{Cr)Ax
+
f(Cz)Ax
+
(C:)Ax
+
...
+(Cn)Ax
=
I f(Cr.)Ax.
,b
rt1
Then,
A-
:
lim
Xf(Cr.)Ax
(this
is not
easyto find)
(sornetimes
impossible).
'(r.r
n-+m
I
-So,
wri
need
a nicq
easy
way to
get
it. That's
where
integration comes into play.
-The
notation f,orthe
Fefuite
rntp,gral ,,
fj$ox.
(the
definite integral
of
(x)
wrr
x
from
a to b)
U
-a
is
the
lower lirtit
of
intsgration,
b is
the upper limit
of
integratio4
&
f(x)
is the
integrand.
-A
definite
Integral
is a
nuqber.
It could
be
positive
,
zera,
or negative.
If may
or
rnay
not represent
en
arca.
fit"
..1,
-Definition:
j
f(x)dx
:
lim
XfICr)Ax
i
(L'
Ue
n-)co
-So,
ifwe
can "$et-up" a
problem
using
this
type of
limit,
we car solve it
be evaluating
the definite
integral
associated
with
this infinite
sum.
*Many
problems
can be
"set-up"
using
the
limit
ofRiemann
Sums. Some
are
problems
involving
area, length
of
curvss,
volume
of solidq surface
area
of solids, motion
of a
particle
on a
curve,
work
done when moving
a
object..
just
to
name
afew. We
will
focus
on the
popular
ones
that
deal with
business applications.
tL
P{gnertiee
of the
Definite
Inteer4!
f.
-l)
{
{*)a*:
o
"ic-
,^'L
-z)
,f
(*)a*
:
-
!f(x)dx
ttd,
\, i/- t+
-3)
Ifa< c
(
b, then.
j
f(x)dx
+
f
f1x;dx
:
/
(*)a*
l'tr.
#s
\)4.
'
'
-a)
If
{x)
>
s(x)
on
[a;b],
then
, {i1*1a,
,.{
t*W"
(used
in
catcurus rr)
utu
*'-
-5)
If M=nas
off(x) and m
=nin
of f(x) over
[qb],
then m(b-a)
<
M(b-a)
