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We are on the cusp of alleviating the tedium of constructing rectangles. Or, as our author says: "Newton and Leibniz...saw that the Fundamental Theorem enabled them to compute areas and integrals...without having to compute them as limits of sums...." (p. 310).
It comes in two parts:
If is continuous on
, then
where is any anti-derivative of
; that is, a function such that
.
In words: definite integration is easy if you know an anti-derivative of the integrand:
is continuous on and differentiable on
, and
.
In words: It's easy to construct an anti-derivative of a function:
This happens to be the particular anti-derivative which is 0 at
:
Ask yourself how you can use the closely-related integral
to solve your problem. How are and
related?