Exercises

In the exercises we will interpret the results of using these commands on particular types of functions.

1. Now let's consider the function on the interval x=0 to x=3. Let A denote the area under the graph of f between x=0 and x=3.
   1. 1. When you use a leftbox command for this function on this interval, what can you say about the ``whitespace'' under the curve and ``greenspace'' above the curve?
      2. What does your answer to (i) tell you about the sum of the area of the rectangles you plot with leftbox in relation to A?
   2. 1. When you use a rightbox command for this function on this interval, what can you say about the ``whitespace'' under the curve and ``greenspace'' above the curve?
      2. What does your answer to (i) tell you about the sum of the area of the rectangles you plot with rightbox in relation to A?
   3. What property of the funtion f made your answers in (a) and (b) turn out the way it did?
   4. For the same number of rectangles, what is the relationship between the numerical results of the leftsum, rightsum, and middlesum commands? Why should this be the case?

2. Repeat Exercise 1 for the function on the interval x=1 to x=3. Let A denote the area under the graph of f between x=1 and x=3.
3. Now let on the interval x=-1 to x=2?
   1. Based upon your answers to Exercises 1 and 2, how might you produce an approximation to the area under this curve on this interval that is less than the actual area? Use the appropriate sum commands to carry out such an approximation.
   2. Based upon your answers to Exercises 1 and 2, how might you produce an approximation to the area under this curve on this interval that is greater than the actual area? Use the appropriate sum commands to carry out such an approximation.
4. Intuitively, we can improve on an approximation to the area under a curve by increasing the number of boxes that we use to produce the approximation. We might also expect that if we could let the number of boxes become infinite, then we could obtain an exact value for the area under a curve. To do this, we must use the limit command in Maple. Here is a sample of the command for the function on the interval .

      limit(value(leftsum( x^2,x=0..2, n)),n=infinity);

   This yields a number, which we expect should be equal to the area under the curve on this interval. To obtain a decimal expansion for this number, use

      evalf(limit(value(leftsum( x^2,x=0..2, n)),n=infinity));

   Consider the function on the interval x=0 to x=3 from Exercise 1.

   1. Based upon your answers in Exercise 1, use the limit command to produce a potentially exact value for A which you know is less than or equal to A. Why do you know this less than or equal to A?
   2. Based upon your answers in Exercise 1, use the limit command to produce a potentially exact value for A which you know is greater than or equal to A. Why do you know this greater than or equal to A?
   3. Based upon your answers to (a) and (b), what is A?