6'4 The Fundamental Theorem of Calculus

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6.4 The Fundamental Theorem of Calculus
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Antiderivatives
Let f (x) be a function. A function F(x) such
that F'(x) f (x) is called the antiderivative
of f.
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The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus - For any
continuous function f with input t, the
derivative of an accumulation function of f is
the function f in terms of x. In symbols, we
write
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The General Antiderivative
If F(x) is the antiderivative of f, then so is
F(x) c where c is an arbitrarily chosen
constant. F(x) c is called the general
antiderivative. We use the following notation to
represent the general antiderivative.
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Antidifferentiation Rules
Antidifferentiation is simply the opposite of
differentiation. The rules for antidifferentiation
are found on page 422. You need to memorize
these.
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Example
Page 426 2, 4, 10, 12, 14, 18, 20, 26, 32
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6.5 The Definite Integral
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The Definite Integral
If f is a continuous function from a to b and F
is the general antiderivative of f, then
The constant c in the general antiderivative will
cancel out. If F(x) c is the general
antiderivative of f, then we have
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Additional Properties
If f is a continuous function from a to b and c
is between a and b, then
This is useful for computing actual area.
If you reverse the limits of integration, you
must change the sign of the integral.
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Example
Page 441 2, 4, 8, 10, 18, 22
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Area of the Region Between Two Curves
If the graph of flies above the graph of g, from
a to b, then the area of the region between the
two curves from a to b is given by
Note If while calculating the area of the region
between two curves you obtain the negative of the
answer you expect, then it is likely that you
have interchanged the positions of the functions
in the integrand.
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Area of the Region Between Two Curves
Similarly, if f and g are two continuous
rate-of-change functions, then the difference
between the accumulated change of f from a to
band the accumulated change of g from a to b is
Note If two rate-of-change functions intersect
in the interval from ato b, then the difference
between their accumulated changes is not the same
as the area of the regions between the two
function. Remember, definite integrals dont
always give you area.
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Example
Page 443 24, 28, 32
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6.6 Average Value and Average Rate of Change
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Average Values
If y f (x) is a continuous piecewise continuous
function describing the quantity from x a to x
b, then the average value of the quantity from
a to b is calculated by using the quantity
function and the formula
The average value has the units the same as the
output of the function f.
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Average Rates of Change
The average rate of change of the quantity, also
called the average value of the rate of change,
can be calculated from the quantity function as
or from the rate-of-change function as
The average rate of change has the same units as
the rate-of-change of f.
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Example
Page 456 2, 4, 12