Midpoint ApproximationTrapezoidal RuleErrorSimpson’s Rule ...
[Pages:44]Midpoint Approximation Trapezoidal Rule Error Simpson's Rule
Midpoint Approximation
Sometimes,
we
need
to
approximate
an
integral
of
the
form
Rb
a
f (x)dx
and
we
cannot find an antiderivative in order to evaluate the integral. Also we may
need
to
evaluate
Rb
a
f (x)dx
where
we
do
not
have
a
formula
for
f (x)
but
we
have data describing a set of values of the function.
Review
We might approximate the given integral using a Riemann sum. Already we
have looked at the left end-point approximation and the right end point
approximation
to
Rb
a
f
(x
)dx
in
Calculus
1.
We
also
looked
at
the
midpoint
approximation M:
Midpoint Rule If f is integrable on [a, b], then
Zb
n
X
f (x)dx Mn = f (x?i )x = x(f (x?1) + f (x?2) + ? ? ? + f (x?n)),
a
i =1
where
b-a
1
x = n
and xi = a+i x and x?i = 2 (xi-1+xi ) = midpoint of [xi-1, xi ].
Annette Pilkington
Approximating an integral
Midpoint Approximation Trapezoidal Rule Error Simpson's Rule
Midpoint Approximation
Example
Use
the
midpoint
rule
with
n
=
6
to
approximate
R4
1
1 x
dx
.
(= ln(4) = 1.386294361)
Fill in the tables below:
Annette Pilkington
Approximating an integral
Midpoint Approximation Trapezoidal Rule Error Simpson's Rule
Midpoint Approximation
Example
Use
the
midpoint
rule
with
n
=
6
to
approximate
R4
1
1 x
dx
.
(= ln(4) = 1.386294361)
Fill in the tables below:
x
=
4-1 6
=
1 2
Annette Pilkington
Approximating an integral
Midpoint Approximation Trapezoidal Rule Error Simpson's Rule
Midpoint Approximation
Example
Use
the
midpoint
rule
with
n
=
6
to
approximate
R4
1
1 x
dx
.
(= ln(4) = 1.386294361)
Fill in the tables below:
x
=
4-1 6
=
1 2
xi x0 = 1 x1 = 3/2 x2 = 2 x3 = 5/2 x4 = 3 x5 = 7/2 x6 = 4
Annette Pilkington
Approximating an integral
Midpoint Approximation Trapezoidal Rule Error Simpson's Rule
Midpoint Approximation
Example
Use
the
midpoint
rule
with
n
=
6
to
approximate
R4
1
1 x
dx
.
(= ln(4) = 1.386294361)
Fill in the tables below:
x
=
4-1 6
=
1 2
xi x0 = 1 x1 = 3/2 x2 = 2 x3 = 5/2 x4 = 3 x5 = 7/2 x6 = 4
x?i
=
1 2
(xi
-1
+ xi )
f (x?i ) =
1 x?i
x?1 = 5/4 4/5
x?2 = 7/4 4/7
x?3 = 9/4 4/9
x?4 = 11/4 4/11
x?5 = 13/4 4/13
x?6 = 15/4 4/15
Annette Pilkington
Approximating an integral
Midpoint Approximation Trapezoidal Rule Error Simpson's Rule
Midpoint Approximation
Example
Use
the
midpoint
rule
with
n
=
6
to
approximate
R4
1
1 x
dx
.
(= ln(4) = 1.386294361)
Fill in the tables below:
x
=
4-1 6
=
1 2
xi x0 = 1 x1 = 3/2 x2 = 2 x3 = 5/2 x4 = 3 x5 = 7/2 x6 = 4
x?i
=
1 2
(xi
-1
+ xi )
f (x?i ) =
1 x?i
x?1 = 5/4 4/5
x?2 = 7/4 4/7
x?3 = 9/4 4/9
x?4 = 11/4 4/11
x?5 = 13/4 4/13
x?6 = 15/4 4/15
M6
=
P6
1
f
(x?i
)x
=
1^4
25
+
4 7
+
4 9
+
4 11
+
4 13
+
4~
15
=
1.376934177
Annette Pilkington
Approximating an integral
Midpoint Approximation Trapezoidal Rule Error Simpson's Rule
Trapezoidal Rule
We
can
also
approximate
a
definite
integral
Rb
a
f
(x
)dx
using
an
approximation
by trapezoids as shown in the picture below for f (x) 0
The
area
of
the
trapezoid
above
the
interval
[xi ,
xi +1 ]
is
x
h
(f
(xi
)+f 2
(xi
+1
)
i .
Trapezoidal Rule If f is integrable on [a, b], then
Zb
x
a f (x)dx Tn = 2 (f (x0) + 2f (x1) + 2f (x2) + ? ? ? + +2f (xn-1) + f (xn))
where
x = b - a and n
Annette Pilkington
xi = a + i x and.
Approximating an integral
Midpoint Approximation Trapezoidal Rule Error Simpson's Rule
Trapezoidal Rule
where
Zb
x
f (x)dx Tn = a
(f (x0) + 2f (x1 ) + 2f (x2 ) + ? ? ? + +2f (xn-1 ) + f (xn )) 2
b-a x =
n
and xi = a + ix and.
Example
Use
the
trapezoidal
rule
with
n
=
6
to
approximate
R4
1
1 x
dx
.
(=
ln(4) = 1.386294361)
Annette Pilkington
Approximating an integral
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