Section Summary: 5.4

  1. Definitions

  2. Theorems

    Total Change Theorem: The integral of a rate of change is the total change:

    displaymath135

    We add up all the small changes in the function F from a to b to discover what the total change was in the function F.

  3. Properties/Tricks/Hints/Etc.

  4. Summary

    The definition of the indefinite integral emphasizes the close relationship between differentiation and integration. These two processes are inverses of each other, in much the same way that the square root is the inverse process of the square.

    The indefinite integral is a function, whereas a definite integral is a number (representing a fixed area). Furthermore, notice that the ``dummy variable of integration'' (x) is used as the variable of the antiderivative, e.g.

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    This is a point of confusion for many students. Remember, however, that ``a rose by any other name would smell as sweet'', and that F(t), F(x), F(nose), and F(rose) all represent the same thing: these variables are just place holders.

Problems:

Problems, pp. 352-355, #1, 2, 8, 13, 26, 41, 46, 54, 62, 63

On the board: #2, 46, 53, 58





LONG ANDREW E
Thu Mar 29 14:06:20 EST 2001