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

b | |

c dx (where c is a constant) | |

0 |

(2) Use this to find F_{1}(h), where

h | ||

F_{1}(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 |

(4) Express your third answer in terms of your second answer, that
is, in terms of F_{1}(h).

(5)-(8) Repeat the above steps, but with the function "x" replacing
the (constant) function "c" and with a corresponding F_{2}(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!