Barehands Integration

(1) Using the definition of the definite integral, calculate

 b c dx   (where c is a constant) 0

(2) Use this to find F1(h), where

 h F1(h)= c dx   (where c is a constant) 0

(3) Use the definition of the definite integral to calculate:

 b c dx   (where c is a constant) a

(5)-(8) Repeat the above steps, but with the function "x" replacing the (constant) function "c" and with a corresponding F2(h).

(9) As a group, make a conjecture about the relationship between a the integral from a to b of a given f(x), and a certain "F(h)" (which you define in terms of f(x)). Explain how your calculations above support your conjecture.

(10) Are there several choices for the "F(x)"? Are they all equally good? Why or why not?

You may work together, as well as individually working in parallel on parts of (1)-(4) and (5)-(8) to save time, combining your work before taking on the last parts of the Activity.

However, be sure that you, and everyone in your group, understands all the "whys" (and pictures) behind each step in your calculation; during this Activity (as well as future Activities over the semester) you should be able to explain to me (not necessarily to be perfect, but to be able to articulate) the reasoning behind each step -- and, if anyone in your group is uncertain about such a question, you should be able to explain to your teammate, to their satisfaction, the reasoning, "whys" and pictures.

Good luck!

©Copyleft 1996 by Harel Barzilai. "Share and Enjoy."