Simpson’s rule: Summary of the last lecture
Sometimes definite integrals cannot be exactly evaluated using known
analytical techniques, and have to be estimated. An example is
Z 4
2
e −x dx.
0
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Numerical integration
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Simpson’s rule: Summary (ctd)
Simpson’s rule is a strategy for estimating a definite integral of the
form
Z b
f (x) dx
a
using the approximation formula:
Sn =
n
∆x X
aj f (xj ),
3
j=0
where n is an even integer, ∆x = (b − a)/n and the coefficients aj
satisfy
(
4 if 1 < j < n, j odd,
a0 = an = 1,
aj =
2 if 1 < j < n, j even.
Math 105 (Section 204)
Numerical integration
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Simpson’s rule: Summary (ctd)
Simpson’s rule is a strategy for estimating a definite integral of the
form
Z b
f (x) dx
a
using the approximation formula:
Sn =
n
∆x X
aj f (xj ),
3
j=0
where n is an even integer, ∆x = (b − a)/n and the coefficients aj
satisfy
(
4 if 1 < j < n, j odd,
a0 = an = 1,
aj =
2 if 1 < j < n, j even.
What is the Simpson’s rule approximation for the example in the first slide?
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Interpetation of Simpson’s rule
Simpson’s rule is exact, i.e., gives the correct answer, for computing areas
under quadratic curves. In other words, the value of any integral of the
form
Z b
2
px + qx + r dx
a
is the same as its Simpson’s rule approximation.
Let’s check this!
Math 105 (Section 204)
Numerical integration
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Explanation of the Simpson’s rule formula
Now fix a general (not necessarily quadratic) curve y = f (x), an interval
[a, b], and a partition
a = x0 < x1 < x2 < · · · < xn = b,
where n is an even integer.
Step 1. Replace the curve y = f (x) on the first subinterval [x0 , x2 ] by a
parabola C1 that agrees with f at three points, namely passes through
(x0 , f (x0 )), (x1 , f (x1 )) and (x2 , f (x2 )). Then
Z
x2
f (x) dx ≈ A1
x0
= area under C1 between [x0 , x2 ]
∆x
[f (x0 ) + 4f (x1 ) + f (x2 )] .
=
3
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Explanation of the Simpson’s rule formula
Step 2. Repeat the same step as above but now on the interval [x2 , x4 ].
Let C2 be the parabola that passes through (x2 , f (x2 )),
(x3 , f (x3 )) and (x4 , f (x4 )).
Z x4
f (x) dx ≈ A2
x2
= area under C2 between [x2 , x4 ]
∆x
=
[f (x2 ) + 4f (x3 ) + f (x4 )] .
3
···
···
···
Continue until you run out of points, in n/2 steps.
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Explanation of the Simpson’s rule formula
Approximate area
Simpson’s rule says that an estimate for the actual area under the curve
y = f (x) on [a, b] is the sum of the area of the approximating parabolas;
namely,
Z
b
f (x) dx ≈ Sn = A1 + A2 + · · ·
a
=
∆x h
f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) + · · · +
3
i
4f (xn−1 ) + f (xn ) .
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How good is the approximation?
Error bound
Assume that f (4) is continuous on the interval [a, b] and that K is a real
number such that
(4) f
(x)
≤ K on [a, b].
Then the error in approximation satisfies the inequality
Z b
K (b − a)
ES = f (x) dx − Sn ≤
(∆x)4 .
180
a
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Example
You program your calculator to compute definite integrals using Simpson’s
rule with n = 6. If you evaluate
Z 6
e −3x dx
0
using your calculator, find the maximum amount by which your result
would differ from the correct answer.
A. 1
B. 81/180
C. 1/180
D. 10/27
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