Problem:
Given the function

f(x) = x3 + 1

approximate the area bounded by the graph of f, the x-axis and the line x = 1.

Visualization:

First, we discuss how to approximate an area using inscribed and circumescribed rectangles.

The sum of the areas of the inscribed rectangles is 1.8 and the sum of the areas of the circumscribed rectangles is 2.2. It should be clear that the actual area bounded by f, the x-axis and the line x = 1 is between these two numbers.

How can we get a better approximation for the area? Let's first look at the following examples:

In both of these examples, we see that by increasing the value of n we obtain a better approximation of the area. Try various values for n and see how close you can get to the actual area bounded by the graphs of f(x) = x3+1, the x-axis and the line x = 1.
Enter a integer value for n:

WARNING If you choose n too large then your browser can "freeze" and you may have to reboot your computer.