Let
f
be a continuous function whose domain includes the closed interval
[a, b]
. We have investigated ways of
approximating the definite integral
We are now interested in determining how good are these approximations. We define the error:
Riemann sums using left-hand endpoints:
Riemann sums using right-hand endpoints:
Riemann sums using midpoints:
Trapezoidal Rule:
Simpson's Rule:
Trapezoidal Rule Error Bound:
Suppose that the second derivative
f''
is continuous on
[a, b]
and suppose that
|f''(x)|
<
M
for all
x
in
[a, b]
. Then
Example #1
[
Using Flash
] [
Using Java
]
[The Trapezoidal Rule approximation was calculated in
Example #1 of this page
.]
Example #2
[
Using Flash
] [
Using Java
]
[The Trapezoidal Rule approximation was calculated on
this page
.]
Example #3
[
Using Flash
] [
Using Java
]
[The Trapezoidal Rule approximation was calculated on
this page
.]
Example #4
[
Using Flash
] [
Using Java
]
Using Maple to find the error bound for the Trapezoidal Rule
.
[The Trapezoidal Rule approximation was calculated on
this page
.]
Simpson's Rule Error Bound:
Suppose that the fourth derivative
f''
is continuous on
[a, b]
and suppose that
|f
^{(4)}
(x)|
<
M
for all
x
in
[a, b]
. Then
Example #5
[
Using Flash
] [
Using Java
]
[The Simpson's Rule approximation was calculated in
Example #2 of this page
.]
Example #6
[
Using Flash
] [
Using Java
]
[The Simpson's Rule approximation was calculated on
this page
.]
Using Maple to find the error bound for Simpson's Rule
.
[The Simpson's Rule approximation was calculated on
this page
.]
Midpoint Rule Error Bound:
Suppose that the second derivative
f''
is continuous on
[a, b]
and suppose that
|f''(x)|
<
M
for all
x
in
[a, b]
. Then