Page 195 - Calculus Workbook For Dummies

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Chapter 10: Integration: Reverse Differentiation

3. In what intervals between 0 and 8 is A g 4. Approximate A 1 ^ h, A 3 ^ h, and A 5 ^ h.
g
g
g
a. increasing?
Solve It
b. decreasing?
Solve It

Sound the Trumpets: The Fundamental

Theorem of Calculus

The absolutely incredibly fantastic Fundamental Theorem of Calculus — some say one
of or perhaps the greatest theorem in the history of mathematics — gives you a neat
shortcut for finding area so you don’t have to deal with the annoying area function or
that rectangle mumbo jumbo from Chapter 9. The basic idea is that you use the anti-
derivative of a function to find the area under it.

3
3
Let me jog your memory on antiderivatives: Because x3 2 is the derivative of x , x is an
2
2
3
3
antiderivative of x3 . But so is x + 5 because its derivative is also x . So anything of the
3
3
2
form x + C (C is a constant) is an antiderivative of x . Technically, you say that x + C
3
2
3
is the indefinite integral of x3 2 and that x + C is the family of antiderivatives of x .
3
The Fundamental Theorem comes in two versions: useless and useful. You learn the
useless version for basically the same reason you studied geometry proofs in high
school, namely, “just because.”