The Area And Distance Problem

# The Area Problem

We are easily able to determine the area of a shape that contains only straight sides. Even in the cases of irregular polygons, we can simply break the shape up into easier shapes to compute their areas. However, sometimes we may need to figure out the area of a shape that does not contain straight sides or any elements related directly to circles or ellipses. How are we able to calculate the area of the shape?

This is the area problem. Suppose we have a function such as:

(1)
$$f(x) = x^2 + x$$

And we want to determine the area on the interval of [0, 2] bounded by the y-axis and the x-axis. This area would be described by the graphic below.

As we clearly see, one of the sides of this shape is not a straight line, hence, it becomes impossible to determine the area of the shape by simplying breaking it up into a series of smaller shapes… or does it?

Suppose we begin this problem by breaking area into a series of rectangles. Let's say we break the function up into 4 rectangles with equal widths along the x-axis, or simply, a width of 0.5 units each. And then let's say the heights are arbitrarily going to match with the function at f(x). Thus we obtain the following table of heights and widths:

 Height Width f(0.5) 0.5 f(1) 0.5 f(3/2) 0.5 f(2) 0.5