4.3 The Definite Integral
Motivating 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?
In Figure 4.3.1, we see evidence that increasing the number of rectangles in a Riemann sum improves the accuracy of the approximation of the net signed area bounded by the given function.
Figure 4.3.1: 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].
We therefore explore the natural idea of allowing the number of rectangles to increase without bound. In an effort to compute the exact net signed area we also consider the differences among left, right, and middle Riemann sums and the different results they generate as the value of n increases. We begin with functions that are exclusively positive on the interval under consideration.
Preview Activity 4.3.1. Consider the applet found at http://gvsu.edu/s/a91. There, you will initially see the situation shown in Figure 4.3.2.
Note that the value of the chosen Riemann sum is displayed next to the word "relative," and that you can change the type of Riemann sum being computed by dragging the point on the slider bar below the phrase "sample point placement."