7.3 - MATH
1081
 Monday,
April
11
 


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Unformatted text preview: MATH
1081
 Monday,
April
11
 
 Chapter
7
–
Section
3
 
 AREA
AND
THE
DEFINITE
INTEGRAL
 Homework
#12
(due
4/18):
 








 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 
 
 Section
7.3
#8,
28,
36
 Section
7.4
#4,
12,
26,
34,
56
 Last
week,
we
found
the
antiderivative
of
a
function.

That
is,
 we
“worked
backwards”
to
figure
out
what
function
we
would
 take
the
derivative
of
to
yield
a
given
function
 f ( x ) .
 
 Today,
we
will
discuss
a
concept
that
is
seemingly
completely
 unrelated
to
the
antiderivative.

However,
as
we
will
see
next
 time,
in
the
Fundamental
Theorem
of
Calculus,
there
is
a
very
 important
connection
between
these
to
concepts.
 
 The
basic
question
for
today
is:
What
is
the
area
of
the
region
 between
the
curve
 f ( x ) 
and
the
 x ‐axis
from
 x = a 
to
 x = b ?
 
 
 12 Let’s
begin
by
looking
at
the
curve
 f ( x ) = − x + 4 
from
 x = 0 
 4 to
 x = 4 
shown
here.
 
 
 Let’s
approximate
the
area
of
the
region
by
using
rectangles,
 for
which
we
know
how
to
find
an
area.

For
example,
if
we
 could
use
the
4
rectangles
shown
here.
 

 
 This
will
clearly
be
an
overestimate
of
the
area
of
the
region,
 but
it
isn’t
off
by
all
that
much.
 
 
 In
this
approximation,
each
rectangle
has
width
1
and
the
 height
is
determined
by
the
height
of
the
curve
at
the
left‐hand
 endpoint
of
the
base.


 
 12 So,
the
height
of
the
first
rectangle
is
 f ( 0 ) = − ( 0 ) + 4 = 4 .
 4 12 15 The
height
of
the
second
rectangle
is
 f (1) = − (1) + 4 = .
 4 4 
 
 
 Rectangle 1: Rectangle 2: Rectangle 3: Rectangle 4: A1 = f ( 0 ) ⋅ 1 = 4 ⋅ 1 = 4 A3 = f ( 2 ) ⋅ 1 = 3 ⋅ 1 = 3 
 A2 = f (1) ⋅ 1 = 15 / 4 ⋅ 1 = 15 / 4 A4 = f ( 3) ⋅ 1 = 7 / 4 ⋅ 1 = 7 / 4 Total Area: 
 TA = A1 + A2 + A3 + A4 = 25 / 2 = 12.5 What
happens
if
we
use
the
right‐hand
endpoint
of
the
base
of
 each
rectangle
to
determine
the
height?
 

 
 
 In
this
case
we
get
an
underestimate
of
the
area
of
the
region
 and
this
approximation
equals
8.5. 1 Example:

Approximate
the
area
under
the
graph
of
 f ( x ) = 

 x 
and
above
the
 x ‐axis
from
 x = 1
to
 x = 3
using
 n = 4 

 
rectangles
(of
equal
width)
and
 
 a. left‐hand
endpoints.
 
 
 
 
 
 b. right‐hand
endpoints.
 
 
 So,
at
this
point,
we
know
that
the
area
of
the
region
 

 
 is
somewhere
between
8.5
and
12.5.

But
how
can
we
get
a
 better
estimate?
 
 Maybe
an
average
of
the
two?
 Maybe
using
the
midpoint
to
determine
the
height?
 
 The
best
way
in
general
to
get
a
better
estimate
is
to
use
more
 rectangles
of
smaller
width.

In
fact,
if
you
use
enough
 rectangles
with
small
enough
widths,
then
you
get
a
good
 estimate
regardless
of
whether
you
use
the
left‐hand
endpoint,
 right‐hand
endpoint,
midpoint,
or
any
other
point
of
the
base
 to
determine
the
height.
 
 To
get
the
best
approximation,
we
return
to
a
concept
of
a
 limit.

In
particular,
if
we
compute
the
limit
of
the
sums
of
the
 areas
of
the
rectangles
as
the
number
of
rectangles
goes
to
 ∞ 
 and
their
widths
go
to
0,
this
is
what
we
define
the
exact
area
 of
the
region
to
be.
 To
be
precise,
if
we
divide
the
interval
[ a, b ]
into
 n 
equal
 b−a subintervals,
each
will
have
width
 = Δx .
 n 
 Then
the
area
can
be
approximated
by
a
sum
that
looks
like
 f ( x1 ) ⋅ Δx + f ( x2 ) ⋅ Δx + f ( x3 ) ⋅ Δx + ... + f ( xn ) ⋅ Δx 
or
 ⎡ f ( x1 ) + f ( x2 ) + f ( x3 ) + ... + f ( xn ) ⎤ Δx .
 ⎣ ⎦ 
 To
compute
the
exact
area,
we
compute
the
limit
 lim ⎡ f ( x1 ) + f ( x2 ) + f ( x3 ) + ... + f ( xn ) ⎤ Δx .
 ⎣ ⎦ 
 n→ ∞ There
is
actually
a
notation
for
such
a
limit.


 It
is
called
a
definite
integral
and
is
written
as
 
 
 
 
 
 
 
 
 So,
geometrically
this
definite
integral
represents
an
area
of
a
 region,
but
it
is
defined
to
be
the
limit
of
a
certain
sum.
 
 
 Recall,
geometrically,
the
derivative
represents
the
slope
of
a
 tangent
line,
but
it
is
define
to
be
the
limit
of
a
certain
quotient. ∫ f ( x ) dx .
 b a Example:

Suppose
that
the
velocity
 v ( t ) 
of
a
vehicle
at
time
 t 

 
is
given
by
the
following
graph.
 
 



 
 
 Approximate
the
total
distance
that
the
vehicle
travelled?
 What
does
the
value
of
 ∫ v ( t ) dt 
tell
us?
 6 1 
Clicker
Check­out:

Choose
any
letter
to
check
out
now.
 
 Next
time
–
Section
7.4
 ...
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This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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