An example from the last lecture
Find the value of the three integrals
Z 3
Z 2
2x
2x
dx
dx
2
2
−2 x − 1
−2 x − 1
Z
2
3
2x
dx.
−1
x2
A. 0, ln(8/3), ln(8/3)
B. 0, ∞, ln(8/3)
C. ∞, ln(8/3), ln(8/3)
D. ∞, ∞, ln(8/3)
Math 105 (Section 204)
Numerical integration
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Numerical integration
Often a definite integral cannot be computed exactly, and its value has to
be estimated with acceptable accuracy.
Math 105 (Section 204)
Numerical integration
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Numerical integration
Often a definite integral cannot be computed exactly, and its value has to
be estimated with acceptable accuracy.
Simpson’s rule
If f is defined and integrable on [a, b], the Simpson’s rule approximation
based on n equally spaced intervals is
Z
b
f (x) dx ≈ Sn =
a
∆x h
f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) · · · +
3
i
· · · + 4f (xn−1 ) + f (xn ) ,
where
n is an even integer,
∆x = (b − a)/n, and
xk = a + k∆x, k = 0, 1, 2, · · · , n.
Math 105 (Section 204)
Numerical integration
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Where does the formula come from?
Find the equation of the parabola
y = px 2 + qx + r
passing through the points (0, 0), (1, a) and (2, b).
A. y = −a +
b
2
x 2 + 2a −
b
2
x
B. y = ax 2 + bx
C. y = bx 2 + ax
D. y = 2a − b2 x 2 + −a + b2 x
Math 105 (Section 204)
Numerical integration
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Where does the formula come from?
Find the equation of the parabola
y = px 2 + qx + r
passing through the points (0, 0), (1, a) and (2, b).
A. y = −a +
b
2
x 2 + 2a −
b
2
x
B. y = ax 2 + bx
C. y = bx 2 + ax
D. y = 2a − b2 x 2 + −a + b2 x
Use the correct answer above to find the area under the parabola between
x = 0 and x = 2. Compare with the Simpson’s rule approximation. They
are equal!
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Numerical integration
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Interpretation of Simpson’s rule
Given a function f on an interval [a, b], consider a partition of [a, b]
into n equal subintervals
a = x0 < x1 < x2 < · · · < xn = b,
and set
Pj = (xj , f (xj )),
j = 0, 1, · · · , n.
Now approximate f by piecewise parabolas. Namely for each of the
following triple of points
(P0 , P1 , P2 ),
(P2 , P3 , P4 ),
(P4 , P5 , P6 ),
· · · , (Pn−2 , Pn−1 , Pn ),
fit a parabola through that set, as in the previous example. In
Simpson’s rule, we approximate the area under f between [a, b] by the
area under this piecewise parabolic curve.
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Numerical integration
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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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Numerical integration
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