APPLICATION OF INTEGRALS

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CHAPTER 5

• APPLICATION OF INTEGRALS

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CONTENT

• 5.1 Area
• 5.1.1 Area Under Curve
• 5.1.2 Area Between Curve
• 5.2 Arc Length
• 5.3 Surface Area
• 5.4 Volume

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OBJECTIVES

• By the end of this chapter student should able
  to
• Calculate area under curve
• Calculate area between two curves
• Calculate arc length
• Calculate surface area
• Calculate volume

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5.1.1 Area Under Curve
If
is a continuous nonnegative function on the
closed interval
, we refer to the area of the region shown in
Figure 5.1
as the area under the graph of
from a to b.
Figure 5.1 Area under a graph
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Area Under Curve

• The computation of the area in Figure 5.1 is not
  trivial matter when the top boundary of the
  region is curved.
• However we can estimate the area to any desired
  degree of accuracy.
• The basic idea is to construct rectangles whose
  total area is approximately the same as the area
  to be computed.
• The area of each rectangle, of course is easy to
  compute.

Figure 5.1 Area under a graph
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Area Under Curve

• Figure 5.2 shows three rectangular approximations
  of the area under a graph.
• When the rectangles are thin, the mismatch
  between the rectangles and the region under the
  graph is quite small.
• In general the rectangular approximation can be
  made as close as desired to the exact area simply
  by making the width of the rectangles
  sufficiently small.

Figure 5.2 Approximating a region with rectangles
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Area Under Curve
on the interval
Given a continuous nonnegative function
Let us divide the x-axis interval into n equal
subintervals, where n represents some positive
integer. Such a subdivision is called a
partition of the interval from a to b. Since
the entire interval is of width
so the width of each of the n subintervals is
Therefore, we could denote this by ?x. That is
(Width of one subinterval)
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Area Under Curve
In each subinterval, select a point. (Any point
in the subinterval will do).
be the point selected from the first
subinterval,
Let
be the point from the second subinterval, and so
on.
These points are use to form rectangles that
approximate the region under the graph of
.
Construct the first rectangle with height
and the first subinterval as base,
as in Figure 5.3.
The top of the rectangle touches
.
the graph directly above
Figure 5.3 rectangles with heights
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Area Under Curve
Notice that
.
The second rectangle rests on the second
subinterval and has height
Thus
Continuing in this way, we construct n
rectangles with a combined area of
(1)
A sum as in (1) is called a Riemann sum. It
provides an approximation to the area under the
graph of
when
is nonnegative and continuous. In fact, as the
number of subintervals increases indefinitely,
the Riemann sums (1) approach a limiting value,
the area under the graph.
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Example 5.1.1
Estimate the area under the graph of the marginal
cost function
from
to
.
Use partitions of 5, 20, and 100 subintervals.
Use the mid points of the subintervals as
to construct the rectangles. See Figure 5.4.
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Example 5.1.1 Solution (i)
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Example 5.1.1 Solution (ii)
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Area Under Curve
is negative at some points in the interval,
In some case which
we may also give a geometric interpretation of
the definite integral.
shown in Figure 5.6.
Consider the function
The figure shows a rectangular approximation of
the region between
the graph and the x-axis from a to b.
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Area Under Curve
.
Consider a typical rectangle located above or
below the selected point
If
is nonnegative, the area of the rectangle equals
.
is negative, the area of the rectangle equals
.
In case
So the expression
equals either the area of the corresponding
rectangle or the negative of the area, according
to whether
is nonnegative or negative, respectively.
In particular, the Riemann Sum
is equal to the area of the rectangles above the
x-axis minus the area of the rectangles below the
x-axis.
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Area Under Curve
This gives us the following geometric
interpretation of the definite integral.
Figure 5.7 Regions above and below the x-axis
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Area Under Curve
Recall the following theorem,
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Area Under Curve
The following summarize the formula for area
under the curve.
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Example 5.1.2
Find the area between x-axis and the graph of f,
from
to
Solution
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Example 5.1.3
Find the area of region between x-axis and the
graph of
from
to
Solution
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Example 5.1.4
Find the area of the region between x-axis and
the graph of
from
to
Solution
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Example 5.1.5
Find the area of the region between x-axis and
the graph of f,
Solution
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EXERCISES 5.1.1

• Find the area between x-axis and the curve

A B C D
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5.1.2 Area between Curves
Lets consider regions that are bounded both above
and below by graphs of functions. Referring to
Figure 5.8, we would like to find a simple
expression for the area of the shaded region


and above the graph of the
under the graph of
from
to

It is the region under the graph of
with the region under the graph of
taken away.
Therefore,

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Area between Curves
Figure 5.8
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Area between Curves
Referring to Figure 5.8, both of the functions
are nonnegative, considering the case where
and
are not always positive.
Let us determine the area of the shaded region in
Figure 5.9(a).
and
Select some constant c such that the graphs of
the functions
lie completely above the x-axis.
Figure 5.9
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Area between Curves
See Figure 5.9(b). The region between them will
have the same area as the original region. Using
the rule as applied to nonnegative functions, we
have
Figure 5.9
and
Therefore, we see that our rule is valid for any
functions
as long as the graph of
lies above the graph of
for all x from
to
.
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Area between Curves

• TIPS
• Draw the curve
• Find where they intersect
• Find the area

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Example 5.1.6
Find the area bounded by the curve
and
between the interval 0, 1.
Solution
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Example 5.1.7
Find the area bounded by
and
Solution
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Example 5.1.8
Find the area between the curve
and
Solution
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Example 5.1.9
Find the area of the region between the curve
and
on 0, 4
Solution
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EXERCISES 5.1.2

• Find the area bounded by the curves

A B C D
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5.2 ARC LENGTH

• In this section, we will use integration to find
  the length of a plane curve.
• The length of a plane curve f (x) over an
  interval a, b is called arc length

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Arc Length

• To avoid some complications, we assume that f
  (x) is continuous on a, b, or in other word, we
  will say that f (x) is a smooth curve on a, b
  or f is a smooth function on a, b.
• Thus we will be concerned with the following
  problem.

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Example 5.2.1
Find the arc length of the curve
to
from
Solution
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Example 5.2.2
Find the arc length of the curve
with
Solution
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EXERCISES 5.2

• Find the arc length of the curve over the state
  interval

A B C
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SURFACE AREA

• In this section, we will focus on rotate a
  section of a curve about a line and consider the
  surface area of the solid created.
• As example, if we take the circle
  and rotate it about the x-axis, we get a sphere.
  The calculation would give us the surface area of
  that sphere.
• In other word, revolving a curve about an axis
  will generate a surface area.

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Revolving a curve about x-axis will generate a
surface area
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Example 5.3.1
Find the area of the surface that is generated by
revolving the portion of the curve
between
about the x-axis
Solution
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Example 5.3.2
The curve
is an arc of the circle
Find the area of the surface obtained by
rotating this arc about the x-axis
Solution
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Revolving a curve about y-axis will generate a
surface area
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Example 5.3.3
Find the area of the surface that is generated by
revolving the portion of the curve
between
and
about the y-axis
Solution
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Example 5.3.4
The arc of the parabola
from (1, 1) to (2, 4) is rotated about the
y-axis.
Find the area of the resulting surface
Solution
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Example 5.3.4
The arc of the parabola
from (1, 1) to (2, 4) is rotated about the
y-axis.
Find the area of the resulting surface
Solution cont
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EXERCISES 5.3

• Find the area of the surface obtained by rotating
  the curve about the x-axis
• The given curve is rotated about y-axis. Find the
  area of the resulting surface

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Thank You