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4.6 The Fundamental Theorem of Calculus
Recall from Section 4.1, if
then:
represents the area under the curve
from
to ,
This formula states that __________ is an antiderivative of _________.
Hence, if
is any other antiderivative of
, then:
Thus,
But we learned in Section 4.5, based on our work with Riemann sums, that we can represent
area as follows:
The Fundamental Theorem of Calculus (Part I)
If is continuous on
and is any antiderivative of
on
, then
(The proof is provided in the textbook (pg. 310) and is left to the interested student to read on
their own.)
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Example 1
Use the Fundamental Theorem of Calculus (FTC) to evaluate the following definite integrals.
The FTC can be applied without modification to definite integrals in which the lower limit of
integration is greater than or equal to the upper limit of integration.
Example 2
Find the area under the curve
over the interval
.
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Warning! If is not continuous on the interval
, then the FTC cannot be applied.
For example, suppose we incorrectly apply the FTC to the function
over the interval
:
Why is this impossible? __________________________________________________________
To integrate a continuous function that is defined piecewise on an interval
, split the
interval into subintervals at the breakpoints of the function, and integrate separately over each
subinterval in accordance with Theorem 4.4.5:
Example 3
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To integrate the absolute value of a function
function.
, represent |
Then integrate the piecewise-defined function as shown previously.
Example 4
Evaluate the integral.
Let’s see what this integral represents graphically!
as a piecewise-defined
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Definition
If is a continuous function on the interval
between the curve
and the interval
, then we define the _____________________
to be:
Example 5
Find the total area between the curve
and the -axis over the interval
.
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The Mean-Value Theorem for Integrals
(The proof is provided in the textbook (pg. 315) and is left to the interested student to read on
their own.)
Example 6
Find all values of in the stated interval that satisfy the Mean-Value Theorem for Integrals,
and explain what these numbers represent.
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If
is the area under the curve
as the definite integral:
over the interval
Differentiating both sides of this equation WRT
, then we can express
yields:
The Fundamental Theorem of Calculus (Part 2)
If is continuous on an interval, then has an antiderivative on that interval. In particular, if
is any point in the interval, then the function defined by
is an antiderivative of ; that is,
notation:
for each
in the interval, or in an alternative
(The proof is provided in the textbook (pg. 317) and is left to the interested student to read on
their own.)
Parts I and II of the Fundamental Theorem of Calculus taken together rank as one of the
greatest discoveries in the history of science, and its formulation by Newton and Leibniz is
generally regarded to be the “discovery of calculus.”
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Example 7
Find the derivative using Part 2 of the Fundamental Theorem of Calculus (FTC). Then check the
result by performing the integration and differentiation.
Example 8
(a)
(b)
(c)
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Part I of the FTC states:
where
is an antiderivative of
. Since
, we may write:
Example 9
(a) If
is the rate of change of the area of an oil spill measured in
the integral
represent, and what are its units?
., what does
(b) If
is the marginal profit (dollars per unit) that results from producing and selling
units of a product, what does the integral
represent, and what are its
units?