One of the early classical mathematical problems of geometry is to find the area of a region S enclosed by a curve. As a special case, consider the problem of finding the area under the graph of y=f(x), above the x-axis, and between x=a and x=b.

Regions which are bounded by polygons [that is, straight lines] can be broken up into unions of rectangles and triangles, the areas of each of which can easily be computed...

What about areas bounded by smooth curves?

If we approximate the area by the union of very small squares, we can approximate the area. In fact, if the squares lie entirely within S, we arrive at a lower bound. If the region S lies entirely within the squares, we have an upper bound. Taking the squares to be smaller and smaller, we arrive (via the squeeze theorem) with not only a definition of the area of the curve, but a way to compute it!

In the special case of the area underneath the graph of a function y=f(x), we can approximate the area by rectangles.

Subdivide the interval [a,b] into sub intervals:

a=x₀ < x₁ < x₂ < ... < x_n-1 < x_n=b and approximate the height on [x_i-1,x_i] by f(x_i^*), where x_i^* is any point in [x_i-1,x_i].

Using the special summation formulas in the previous section, we can calculate the area underneath polynomials, xⁿ.

In maple 5.3 or 5.4, we have:

with(student); (to load the student package)
leftsum(x^2, x=0..3, 12);
middlesum(x^2, x=0..3, 12);
rightsum(x^2, x=0..3, 12);

Graphically, Maple 5.3 or 5.4 has the following very useful commands:

with(student); (to load the student package)
leftbox(x^2, x=0..3, 12);
middlebox(x^2, x=0..3, 12);
rightbox(x^2, x=0..3, 12);

Relying on closed form summation formulas, and limits to calculate areas underneath general functions is quite difficult, and we seek better ways to calculate areas.

Last modified Wed Nov 6 23:04:49 CST 1996