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Файл:Mathematics for Foreign Students integrals. Study Guide
.pdf
61
A P P L I C A T I O N S O F D E F I N I T E I N T E G R A L S
There are many applications of definite integrals, such as calculating
planar areas, arc length of a function, finding volumes of revolution solids,
finding average centers, moment of inertia for planes and revolution solids,
in problems of movement of solids, finding liquid pressure, as well as in calculating the velocity of gas molecules and many other applications.
Here we will look at the plane areas, the length of the arc, and the volumes of revolution solids.
The area of a region is defined by the curve of function
and the x-axis in the interval
.
Example. Find the area of the region is defined by the function curve
, the x-axis, and the lines
Solution:
Figure 1
62
From the figure (1), shows that so the area
is given by the following
Example. Find the area of the region which is defined by the function
curve and the X-axis.
Solution:
Figure 2
We find the x-coordinates of the points of intersection between
the function and the x-axis by making , we find that
so that
We are looking at whether or , on the interval
, we find that , so the area is given by the fol-
lowing
63
Example. Find the area of the region is defined by the function curve
and the -axis, in the interval
.
Solution:
Figure 3
We find the x-coordinates of the common points between the function
and the x-axis by making , we find that
64
The curve intersects the x-axis at the point , so the area is given
by the following
Example. Find the area of the region which is defined by the function
curve and the X-axis, in the interval
.
Solution:
Figure 4
65
From the figure 4, shows that the curve intersects the x-axis at the point
, so the area is given by the following
2
The area of a region is defined by two curves of functions
in the interval
Example. Find the area of the region is defined by the curves of func-
tions , and the lines
Solution:
Figure 5
We see that , so the area is given by the
following
66
Example. Find the area of the region which is defined by the curves
of functions , and the lines
Solution:
The curves do not intersect, so we take an optional value from the in-
terval and let it be .
.
From the above, the area of the region is given by the following
Example. Find the area of the region which is defined by the parabola
and the line .
Solution:
Figure 6
67
To find the x-coordinates of the two points of intersection, we make
, we get , so that
The limits of integration are , we take an optional value for the in-
terval and let it be .
.
From the above, the area of the region is given by the following
Example. Find the area of the region is defined by the curves of func-
tions .
Solution:
Figure 7
68
To find the x-coordinates of the intersection’s points of the two curves,
we make , we find that
so that
The limits of integration are , so the area is given by
the following
Example. Find the area of the region which is defined by the curves
of functions .
Solution:
Figure 8
69
To find the x-coordinates of the intersection’s points of the two curves,
we make , we get
so that
From above the integration will be in two intervals
,
so the area is given by the following
Example. Find the area of the region which is defined by the curves
of functions
.
Solution:
Figure 9
70
To find the x-coordinates of the points of intersection between the two
curves, we make , we get
so that
From above the integration will be in two intervals
, so the area is given by the following
=
Area in polar coordinate
Consider the region which is bounded by a polar curve
and two semi-straight lines and
Figure 10
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