||One basic problem we have been exploring in this chapter is how to compute the distance traveled from a given velocity function. We examined this in terms of antiderivatives in section 4.1 and reworked this as an area problem in section 4.2. In this section, we will develop the general problem of calculation of areas in some detail.
You are all familiar with the formulas for computing the area of a rectangle, a circle and a triangle. From endless use of these formulas over the years, you probably have good intuition about what area is: one measure of the size of a two-dimensional region. But, how can we compute the area of a region that's not a rectangle, circle or triangle? Notice that we've used the word compute. For most regions, areas are not measured directly, but rather are computed using some one-dimensional measurements and a formula.
The problem of computing areas is more important than you might think. There are fairly obvious examples such as the area of your front yard (how much grass seed is needed?) and the area of the living room floor (how much carpet is needed?). We saw in section 4.2 that areas are also computed to determine the distance traveled. We see later how to interpret area to give the probability of a random event or the volume of oil flowing through a pipeline.
What we need, then, is a more general description of area, one that can be used to find the area of almost any two-dimensional region imaginable. In this section, we develop a general process for computing area. It turns out that this process (which we generalize to the notion of the definite integral in section 4.4) has significance far beyond the calculation of area. In fact, this powerful and flexible tool is one of the central ideas of calculus, with applications in a wide variety of fields.
Let's start our investigation of area by looking at a fairly simple example: we'll try to compute the area bounded between the graph of y = 2x-2x2 and the x - axis. Since 2x-2x2 = 0 for x = 0 and x = 1 , the region we are interested in extends from x = 0 to x = 1 (see Figure 4.5).
Area under 2x-2x2.
The region is clearly not a rectangle, circle or triangle, so we have no handy formula available for computing the area. Since we don't know how to compute the area exactly, we first try to approximate it, using what we already know about area. We'll then suggest a means for systematically improving the approximation.
Although we often think of graphing calculators and computers as drawing curves, what they actually do is plot points, usually by coloring in small squares on the screen called pixels. If you look closely enough at a calculator display of the graph of y = 2x-2x2 , you will probably see a picture similar to Figure 4.6.
(Computer screens look the same way, but their resolution is so fine as to make individual pixels indistinguishable to the naked eye.) In the case of Figure 4.6, each pixel represents a square of side 0.1. To estimate the area, then, we simply count the number of pixels in the region of interest and multiply the total by 0.01, the area of 1 pixel. Try this now: count the highlighted pixels forming the curve and the pixels underneath, but not the pixels forming the x - axis. You should find 33 pixels, yielding an approximate area of 0.33. We can simplify the counting and gain some valuable insight by rethinking the problem somewhat. In Figure 4.7, we have organized the pixels we want to count into columns: the first column is 2 pixels high, the second column is 3 pixels high and so on.
Calculator display of y = 2x-2x2.
Notice that the area indicated in Figure 4.5 can be approximated by the sum of the areas of the rectangles indicated in Figure 4.7. Each rectangle has width 0.1 (1 pixel wide) but the heights differ. What determines the height of each rectangle? Since the top pixel is supposed to represent the graph of y = f (x) , the height is determined by the function values. For example, we have f (0.1) = 0.18 and 0.18 rounds off to 0.2 or two pixels, so that the first column is 2 pixels high. Similarly, f (0.2) = 0.32 and so, the second column is 3 pixels high. Ignoring the rounding off, each
rectangle has width 0.1 and height f (x) , for some x. The total area is then
Counting pixels to approximate area.
After factoring out the common width of 0.1 and evaluating the function, we get
| A = 0.1f (0.1)+0.1f (0.2)+0.1f (0.3)+ +0.1f (0.9)+0.1f (1.0). |
We will see later that the exact area in Figure 4.5 is 1/3 , so we have found a fairly good approximation.
| A = (0.18+0.32+0.42+0.48+0.5+0.48+0.42+0.32+0.18+0)(0.1) = 0.33. |