5.6. NUMERICAL INTEGRATION
319
5.6 Numerical Integration
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How do we accurately evaluate a definite integral such as
1
0
e −x 2 dx when we
cannot use the First Fundamental Theorem of Calculus because the integrand
lacks an elementary algebraic antiderivative? Are there ways to generate accurate
estimates without using extremely large values of n in Riemann sums?
• What is the Trapezoid Rule, and how is it related to left, right, and middle Riemann
sums?
• How are the errors in the Trapezoid Rule and Midpoint Rule related, and how can
they be used to develop an even more accurate rule?
Introduction
When we were first exploring the problem of finding the net-signed area bounded by a
curve, we developed the concept of a Riemann sum as a helpful estimation tool and a
key step in the definition of the definite integral. In particular, as we found in Section 4.2,
recall that the left, right, and middle Riemann sums of a function f on an interval [a, b]
are denoted L n , R n , and M n , with formulas
L n = f (x 0 )△x + f (x 1 )△x + · · · + f (x n−1 )△x =
n−1
i=0
f (x i )△x,
(5.11)
R n = f (x 1 )△x + f (x 2 )△x + · · · + f (x n )△x =
n
i=1
f (x i )△x,
(5.12)
M n = f (x 1 )△x + f (x 2 )△x + · · · + f (x n )△x =
n
i=1
f (x i )△x,
(5.13)
where x 0 = a, x i = a + i△x, x n = b, and △x =
b−a
n . For the middle sum, note that
x i = (x i−1 + x i )/2.
Further, recall that a Riemann sum is essentially a sum of (possibly signed) areas of
rectangles, and that the value of n determines the number of rectangles, while our choice
of left endpoints, right endpoints, or midpoints determines how we use the given function
to find the heights of the respective rectangles we choose to use. Visually, we can see the
similarities and differences among these three options in Figure 5.14, where we consider
the function f (x) =
1
20 (x − 4) 3 + 7 on the interval [1, 8], and use 5 rectangles for each of
319
5.6 Numerical Integration
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How do we accurately evaluate a definite integral such as
1
0
e −x 2 dx when we
cannot use the First Fundamental Theorem of Calculus because the integrand
lacks an elementary algebraic antiderivative? Are there ways to generate accurate
estimates without using extremely large values of n in Riemann sums?
• What is the Trapezoid Rule, and how is it related to left, right, and middle Riemann
sums?
• How are the errors in the Trapezoid Rule and Midpoint Rule related, and how can
they be used to develop an even more accurate rule?
Introduction
When we were first exploring the problem of finding the net-signed area bounded by a
curve, we developed the concept of a Riemann sum as a helpful estimation tool and a
key step in the definition of the definite integral. In particular, as we found in Section 4.2,
recall that the left, right, and middle Riemann sums of a function f on an interval [a, b]
are denoted L n , R n , and M n , with formulas
L n = f (x 0 )△x + f (x 1 )△x + · · · + f (x n−1 )△x =
n−1
i=0
f (x i )△x,
(5.11)
R n = f (x 1 )△x + f (x 2 )△x + · · · + f (x n )△x =
n
i=1
f (x i )△x,
(5.12)
M n = f (x 1 )△x + f (x 2 )△x + · · · + f (x n )△x =
n
i=1
f (x i )△x,
(5.13)
where x 0 = a, x i = a + i△x, x n = b, and △x =
b−a
n . For the middle sum, note that
x i = (x i−1 + x i )/2.
Further, recall that a Riemann sum is essentially a sum of (possibly signed) areas of
rectangles, and that the value of n determines the number of rectangles, while our choice
of left endpoints, right endpoints, or midpoints determines how we use the given function
to find the heights of the respective rectangles we choose to use. Visually, we can see the
similarities and differences among these three options in Figure 5.14, where we consider
the function f (x) =
1
20 (x − 4) 3 + 7 on the interval [1, 8], and use 5 rectangles for each of