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4.5 The Definite Integral
Suppose is a continuous function defined on the interval
negative values.
Net Signed Area
, but that
attains positive and
[Area of regions above the -axis] [Area of regions below the -axis]
Example 1
Indicate the sign of each of the net areas shown:
Sign:_________________
Example 2
Find the net signed area
Sign:_________________
between
and the interval
Area
Area
Area
Sign:_________________
.
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Recall the definition of area from Section 4.4:
Up until now, we have assumed that the rectangles formed in the definition of area under the
curve from to have had equal width
.
What if we allow the widths of the rectangles to vary? A partition of the interval
collection of points
that divides
into
is a
subintervals (possibly of unequal widths):
Then the width of the th rectangle is given by ____________________________.
The sum of the areas of the
follows:
rectangles using this general partition is thus represented as
This sum is called a _______________________________________.
The largest of the widths of the rectangles is denoted by ________________ and is called the
mesh size of the partition.
Example 3
Find the values of:
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Net signed area,
Theorem 4.5.2
If a function is __________________________ on an interval
, then
the _________________________________________between the graph of
and the interval
is given by:
*When this limit exists and does not depend on the choice of partitions or on the choice of the
points
in the subintervals, the function
is called integrable. All continuous functions
are integrable.
Example 4
Express the following limit as an integral. (Do not evaluate the integrals.)
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Example 5
Express the integral as the limit of a Riemann sum. (Do not evaluate the integral.)
Example 6
Sketch the region whose signed area is represented by the definite integral, and evaluate the
integral using appropriate formulas from geometry.
Properties of the Definite Integral (Definition 4.5.3/Theorem 4.5.4)
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Example 7
Use properties of the definite integral (Theorem 4.5.4) and appropriate formulas from
geometry to evaluate the integral.
Theorem 4.5.5
If is integrable on a closed interval containing the three points
and , then
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Example 8
Theorem 4.5.6
Suppose and
are integrable on
.
(a) If
for all
in
, then
(b) If
for all
in
, then
(c) If
for all
in
, then
Example 9
Use Theorem 4.5.6 to determine whether the value of the integral is positive or negative.