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4.3. THE DEFINITE INTEGRAL
4.3 The Definite Integral
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How does increasing the number of subintervals affect the accuracy of the approximation generated by a Riemann sum?
• What is the definition of the definite integral of a function f over the interval
[a, b]?
• What does the definite integral measure exactly, and what are some of the key
properties of the definite integral?
Introduction
In Figure 4.17, which is repeated below as Figure 4.19, we see visual evidence that increasing
the number of rectangles in a Riemann sum improves the accuracy of the approximation
of the net signed area that is bounded by the given function on the interval under
consideration. We thus explore the natural idea of allowing the number of rectangles to
y = f (x)
a
b
c
d
y = f (x)
a
b
c
d
y = f (x)
A 1
A 2
A 3
a
b
c
d
Figure 4.19: At left and center, two left Riemann sums for a function f that is sometimes
negative; at right, the exact areas bounded by f on the interval [a, d].
increase without bound in an effort to compute the exact net signed area bounded by a
function on an interval. In addition, it is important to think about the differences among
left, right, and middle Riemann sums and the different results they generate as the value
of n increases. As we have done throughout our investigations with area, we begin with
functions that are exclusively positive on the interval under consideration.