326
5.6. NUMERICAL INTEGRATION
measures whether the rule generates an over- or under-estimate) tied to the rule used and
the function’s concavity, but the magnitude of the errors generated by T n and M n seems
closely connected. In particular, the errors generated by the Midpoint Rule seem to be
about half the size of those generated by the Trapezoid Rule.
That is, we can observe in both examples that E M,4 ≈ −
1
2 E T,4 and E M,8 ≈ −
1
2 E T,8 ,
which demonstrates a property of the Midpoint and Trapezoid Rules that turns out to hold
in general: for a function of consistent concavity, the error in the Midpoint Rule has the
opposite sign and approximately half the magnitude of the error of the Trapezoid Rule.
Said symbolically,
E M,n ≈ −
1
2
E T,n .
This important relationship suggests a way to combine the Midpoint and Trapezoid Rules
to create an even more accurate approximation to a definite integral.
Simpson’s Rule
When we first developed the Trapezoid Rule, we observed that it can equivalently be
viewed as resulting from the average of the Left and Right Riemann sums:
T n =
1
2
(L n + R n ).
Whenever a function is always increasing or always decreasing on the interval [a, b],
one of L n and R n will over-estimate the true value of
b
a
f (x) dx, while the other will
under-estimate the integral. Said differently, the errors found in L n and R n will have
opposite signs; thus, averaging L n and R n eliminates a considerable amount of the error
present in the respective approximations. In a similar way, it makes sense to think about
averaging M n and T n in order to generate a still more accurate approximation.
At the same time, we’ve just observed that M n is typically about twice as accurate as
T n . Thus, we instead choose to use the weighted average
S 2n =
2M n + T n
3
.
(5.15)
The rule for S 2n giving by Equation 5.15 is usually known as Simpson’s Rule. 9 Note that we
use “S 2n ” rather that “S n ” since the n points the Midpoint Rule uses are different from the
n points the Trapezoid Rule uses, and thus Simpson’s Rule is using 2n points at which to
evaluate the function. We build upon the results in Table 5.1 to see the approximations
generated by Simpson’s Rule. In particular, in Table 5.2, we include all of the results in
9 Thomas Simpson was an 18th century mathematician; his idea was to extend the Trapezoid rule, but
rather than using straight lines to build trapezoids, to use quadratic functions to build regions whose area was
bounded by parabolas (whose areas he could find exactly). Simpson’s Rule is often developed from the more
sophisticated perspective of using interpolation by quadratic functions.
5.6. NUMERICAL INTEGRATION
measures whether the rule generates an over- or under-estimate) tied to the rule used and
the function’s concavity, but the magnitude of the errors generated by T n and M n seems
closely connected. In particular, the errors generated by the Midpoint Rule seem to be
about half the size of those generated by the Trapezoid Rule.
That is, we can observe in both examples that E M,4 ≈ −
1
2 E T,4 and E M,8 ≈ −
1
2 E T,8 ,
which demonstrates a property of the Midpoint and Trapezoid Rules that turns out to hold
in general: for a function of consistent concavity, the error in the Midpoint Rule has the
opposite sign and approximately half the magnitude of the error of the Trapezoid Rule.
Said symbolically,
E M,n ≈ −
1
2
E T,n .
This important relationship suggests a way to combine the Midpoint and Trapezoid Rules
to create an even more accurate approximation to a definite integral.
Simpson’s Rule
When we first developed the Trapezoid Rule, we observed that it can equivalently be
viewed as resulting from the average of the Left and Right Riemann sums:
T n =
1
2
(L n + R n ).
Whenever a function is always increasing or always decreasing on the interval [a, b],
one of L n and R n will over-estimate the true value of
b
a
f (x) dx, while the other will
under-estimate the integral. Said differently, the errors found in L n and R n will have
opposite signs; thus, averaging L n and R n eliminates a considerable amount of the error
present in the respective approximations. In a similar way, it makes sense to think about
averaging M n and T n in order to generate a still more accurate approximation.
At the same time, we’ve just observed that M n is typically about twice as accurate as
T n . Thus, we instead choose to use the weighted average
S 2n =
2M n + T n
3
.
(5.15)
The rule for S 2n giving by Equation 5.15 is usually known as Simpson’s Rule. 9 Note that we
use “S 2n ” rather that “S n ” since the n points the Midpoint Rule uses are different from the
n points the Trapezoid Rule uses, and thus Simpson’s Rule is using 2n points at which to
evaluate the function. We build upon the results in Table 5.1 to see the approximations
generated by Simpson’s Rule. In particular, in Table 5.2, we include all of the results in
9 Thomas Simpson was an 18th century mathematician; his idea was to extend the Trapezoid rule, but
rather than using straight lines to build trapezoids, to use quadratic functions to build regions whose area was
bounded by parabolas (whose areas he could find exactly). Simpson’s Rule is often developed from the more
sophisticated perspective of using interpolation by quadratic functions.