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The Fundamental Theorem of Calculus
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"The Fundamental Theorem of Calculus
Through the examples we looked at for the area
under graphs of functions, we were led to an interesting
observation: there seems to be a relationship between the process of
integration, which is just a fancy way of performing sums, and the
process of differentiation. In fact, this observation is the one
basic fact which underlies almost all of our work in this course.
Consequently, we will give it a name which indicates its importance:
The Fundamental Theorem of Calculus.
What does the Fundamental Theorem mean?
Before we jump in and tell you about the theorem, we will try and give
you an intuitive feel for it through a demonstration. We have seen
already that the definite integral of a positive function can be
interpreted as the area under the graph of the function.
But what about functions which are negative? There's a pretty
simple explanation in that case as well. Remember that the definite
integral is given by a sum
where the points
are formed by breaking the
interval
into n pieces of width
. When the function f was
positive, we could interpret each term
as
the area of a very thin rectangle. However, remember that area is
always considered to be positive and so if the function
f is negative, the term
represents"
....
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