MAT 1235 Calculus II Section 7.7 Approximate (Numerical) Integration .
Post on 26-Dec-2015
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Idea
Many useful Integrals are difficult/impossible to evaluate by the Fundamental Theorem of Calculus
21 1
2
0
xe dx
Facts
In the last lab, we learn different methods of estimating the value of definite integrals.
In general, given a method, bigger n gives better approximation.
We want to find the smallest n such that
|Error|<certain accuracy Why?
Facts
In the last lab, we learn different methods of estimating the value of definite integrals.
In general, given a method, bigger n gives better approximation.
We want to find the smallest n such that
|Error|<certain accuracyIn general, we do not know the error. Why?
Rules Midpoint Rule
• Rectangles
• Height of rectangle = function value of the midpoint of each subinterval
Trapezoidal Rule• Trapezoids
Simpson’s Rule• Parabolas
• n = even
Error Bounds - Trapezoidal Rule
3
2
Consider the integral ( )
Suppose for
Let error in the Trapezoidal Rule
Then,
( )
1
(
2
)
b
a
T
T
f x K a x b
f x dx
E
K b aE
n
Error Bounds - Trapezoidal Rule
3
2
Consider the integral ( )
Suppose for
Let error in the Trapezoidal Rule
Then,
( )
1
(
2
)
b
a
T
T
f x K a x b
f x dx
E
K b aE
n
??????
Example 1
How large should we take n in order to guarantee that the trapezoidal rule approximation for
is accurate to within 0.0001?
2
3
12
)(
n
abKET
2
12
1dx
x
Example 1: Analysis
We want to find the smallest n such that
If so, then
2
12
1dx
x ?? ,1
)( ,2 ,12
Kx
xfba
2
3
12
)(
n
abKET
Example 1: Find K 2
3
12
)(
n
abKET
2
1 ( ) , ??f x K
x
( ) for 1 2f x K x
We need to find the abs. max. of
on [1,2]
( )f x
Modified Closed Interval Method
Abs. max. of |h(x)| occurs at the end points or critical numbers of h(x).
( )h x
x
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