Summary

As if by total surprise, we begin considering the problem of calculating an area. While this may seem like a disappointing end for those of you going on to study business, you should be aware that this area problem is just the easiest and most intuitive example of integration. Lots of business calculations can be thought of as ``areas under a curve'', so never fear! This really is practical....

Example: #24, p. 413

As you were warned, the indefinite integral will turn into a definite integral, when we put limits on the integral sign:

The Definite Integral: If f is defined on the interval [a,b], the definite integral of f from a to b is given by

displaymath155

provided the limit exists, where tex2html_wrap_inline169 and tex2html_wrap_inline171 is any value of x in the tex2html_wrap_inline175 interval. a is called the lower limit of integration, and b is called the upper limit of integration.

Example: #3, p. 411

You can think of the sum on the right-hand side as representing little rectangles, whose

  1. locations along the x-axis are given by tex2html_wrap_inline171 ,
  2. heights are given by tex2html_wrap_inline185 , and
  3. widths are given by tex2html_wrap_inline187 .
Now you see that the dx in the integral is not merely indicating the variable of integration, but has a direct analogy with distance - it really is important! It's playing the role of the tex2html_wrap_inline187 in the sum, so it represents a width of a rectangle. f(x) is also representing a height: it's just that we're computing an infinite number of wafer thin (actually thinner!) rectangles, located at x of height f(x) and width dx, as x varies from a to b.

The definite integral can be approximated by

displaymath156

and will be when your calculator can't figure out how to compute the solution exactly, using formulas from calculus. We'll talk next time about the Fundamental Theorem of Calculus, which tells us how to compute these exactly (when we can!).

Often the choice of the locations tex2html_wrap_inline171 is made according to some scheme. Several popular schemes are

  1. left-endpoints,
  2. right-endpoints, and
  3. mid-endpoints.
Each has its advantages. Mid-points tends to give a more accurate approximation.

Example: #9, p. 412

Example: #15, p. 412



Fri Dec 3 08:32:54 EST 2004