Suppose is a non-negative, continuous function defined on some interval . Then by the area under the curve between and we mean the area of the region bounded above by the graph of , below by the axis, on the left by the vertical line , and on the right by the vertical line .
All of the numerical methods in this lab depend on subdividing the interval into subintervals of uniform length. For example, dividing the interval into four uniform pieces produces the subintervals , , , and .
The Maple student package has commands for visualizing these three rectangular area approximations. To use them, you first must load the package via the with command. Then try the three commands given below. Make sure you understand the differences between the three different rectangular approximations. Take a moment to see that the different rules choose rectangles which in each case will either underestimate or overestimate the area.
There are also Maple commands leftsum, rightsum, and middlesum to sum the areas of the rectangles, see the examples below. Note the use of evalf to obtain numerical answers.
It should be clear from the graphs that adding up the areas of the rectangles only approximates the area under the curve. However, by increasing the number of subintervals the accuracy of the approximation can be increased. All of the Maple commands described so far in this lab permit a third argument to specify the number of subintervals. The default is 4 subintervals. See the example below for appoximations to the area under from to using the rightsum command with 4, 10, 20 and 100 subintervals. (As this region describes a right triangle with height 2 and base 2, this area can be easily calculated to be exactly 2.) Try it yourself with the leftsum and middlesum commands.
Since, in this trivial example, we knew that the area is exactly 2, it appears that, as the number of subintervals increases, the rectangular approximation becomes more accurate. What would happen with a not so trivial region? The next section describes a way of assessing the accuracy of the midpoint rule.
where is the absolute maximum of on , assuming that is continuous on that interval. In practice, is often approximated by a number that is an upper bound for , that is . For instance, if on , might be taken as 4. Do you see why? For more complicated functions, Maple can be used to get a value for that is close to or actually equal to . Note that the error bound formula gives a worst case estimate, the accuracy achieved for a given number of subintervals may be much better than the guarantee given by the formula.