Numerical Methods of Integration
For some functions it is impossibly to find a closed form (analytical) antiderivative. For these functions, we can approximate the area underneath
them by using numerical methods. We’ve already done this using left-hand
and right-hand Riemann Sums. These, however, give fairly bad approximations in general and require a large increase in the number of subintervals for
a relatively small increase in accuracy.
For example. The following elementary functions do not have antiderivatives which are elementary functions.
√
√
3
x 1−x
√
x cos(x)
cos(x)
x
√
1 − x3
sin(x2 )
That is, the Fundamental Theorem of Calculus cannot be applied to these
functions, so to evaluate a definite integral having on of these (or similar)
functions as integrands we would need to use the approximation methods in
this section.
Objectives
• Approximate a definite integral using the Midpoint Rule
• Approximate a definite integral using the Trapezoidal Rules
• Approximate a definite integral using Simpson’s Rule
• Analyze the approximation error in the Trapezoidal Rule and Simpson’s
Rule
The Midpoint Rule
Definition 1 (Midpoint Rule)
Rb
a
f (x)dx ≈ M ID(n), where
M ID(n) =
n
X
f (xi )∆x
i=1
∆x =
b−a
n
and xi is the midpoint of the ith subinterval.
1
Example 1 Approximate
Rπ
0
sin(x)dx using M ID(4).
Solution. Since we are using M ID(4), we know ∆x =
approximation looks like the following:
π−0
4
= π4 . And our
f(x)
π/4
π/2
3π/4
x
π sin(x)
Notice that we have four rectangles, and the total area is as follows:
π−0
π
3π
5π
7π
M ID(4) =
sin( ) + sin( ) + sin( ) + sin( )
4
8
8
8
8
≈
pi
[2.6131]
4
≈ 2.0523
The Trapezoidal Rule
Definition 2 (Trapezoidal Rule)
T RAP (n) =
Rb
a
f (x)dx ≈ T RAP (n), where
LEF T (n) + RIGHT (n)
2
and LEF T (n) and RIGHT (n) are the left-hand and right-hand Riemann
Sums with n subintervals.
Example 2 Approximate
Rπ
0
sin(x)dx using T RAP (4).
2
Solution. Since we are using T RAP (4), we know ∆x =
approximation looks like the following:
π−0
4
= π4 . And our
f(x)
π/4
π/2
3π/4
x
π sin(x)
Notice that we have four rectangles, and the total area is as follows:
(sin(0)+sin( π4 )+sin( π2 )+sin( 3π4 ))+(sin( π4 )+sin( π2 )+sin( 3π4 )+sin(π))
π−0
T RAP (4) = 4
2
pi 4.8284
≈
4
2
≈ 1.8961
Simpson’s Rule
Definition 3 (Simpson’s Rule)
SIM P (n) =
Rb
a
f (x)dx ≈ SIM P (n), where
2M ID(n) + T RAP (n)
3
for M ID(n) and T RAP (n) as above.
3
Example 3 Approximate
Rπ
0
sin(x)dx using SIM P (4).
Solution. By what we have done previously,
SIM P (4) =
=
2M ID(4) + T RAP (4)
3
2(2.0523) + 1.8961
3
≈ 2.00023
Error Estimates
Theorem 1 Error Estimates for the Trapezoidal Rule and Simpson’s Rule.
• If f has a continuous second derivative on [a, b], then the error ET in
Rb
approximating a f (x)dx by the Trapezoidal Rule is
ET ≤
(b − a)3
[max |f 00 (x)|] ,
12n2
a≤x≤b
• Moreover, if f has a continuous fourth derivative on [a, b], then the
Rb
error ES in approximating a f (x)dx by Simpson’s Rule is
ES ≤
(b − a)5 (4)
max
|f
(x)|
,
180n4
4
a≤x≤b
Example 4 Bound the Error in approximating
and SIM P (4).
Rπ
0
sin(x)dx by T RAP (4)
Solution.
T RAP (4): Let f (x) = sin(x). Then
f 0 (x) = cos(x)
f 00 (x) = − sin(x)
We need to maximize f 00 (x), so we find critical points by looking at
f (3) (x)
f (3) (x) = − cos(x)
Critical Points:
− cos(x) = 0
=⇒ x = π/2
Therefore we have three points to check, 0, π/2, and π.
f 00 (0) = 0
f 00 (π/2) = −1
f 00 (π) = 0
This gives that max |f 00 (x)| = 1 on the interval [a, b]. So, we have
ET ≤
(π − 0)3
·1
12(4)2
=⇒ ET ≤
π3
≈ .16149
192
SIM P (4): Similarly to the error bound for T RAP (4), we find critical points for
the fourth derivative of sin(x)
f (4) (x) = sin(x)
f (5) (x) = cos(x)
Sritical Points:
cos(x) = 0
=⇒ x = π/2
5
Again, we have three points to consider 0, π/2, and π.
f (4) (0) = 0
f (4) (π/2) = −1
f (4) (π) = 0
This gives that max |f (4) (x)| = 1 on the interval [a, b]. So, we have
(π − 0)5
ES ≤
·1
180(4)4
=⇒ ES ≤
6
π5
≈ .00664
46080