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Файл:Mathematics for Foreign Students integrals. Study Guide
.pdf
71
The area of the polar region is given by
Example. Find the area is enclosed by the cardioid .
Solution:
Figure 11
We can find the area of cardioid by integrating the polar equation
in the interval , as a following
72
Area of a region is bounded by a parametric curve
Suppose that the curve is defined in parametric form by the equations
and if the parameter runs between
. Then
the area under the curve is given by the formula,
Example. Find the area under the curve of the cycloid which is defined
by the equations
in the interval
Solution:
Figure 12
=
73
Arc length
If and are two points on the curve , where
and its derivative are continuous on the interval , the length
of arc is given by
Similarly, if and are two points on the curve ,
where and its derivative are continuous on the interval ,
the length of arc is given by
Suppose that the curve is defined in parametric form by the equations
and if the parameter runs between
. And if
and
, is given by the formula
If a curve is given in polar coordinates , hence the length
of arc from to
, is given by the formula
Example. Find length of arc of the curve
, in the interval
74
Solution:
so that
, then the length of arc is given by
following
=
.
Figure 13
Example. Find length of arc of the curve
, in
the interval
75
Solution:
Figure 14
so that
, then the length of arc is
given by following
=
Example. Find the length of arc of the cycloid which is defined by the equa-
tions in the interval
Solution:
Figure 15
76
We have
, and
then the length of arc is given by following
=
.
Example. Find the length of arc of the curve
in the inter-
val
Solution:
Figure 16
77
We have
, and
then the length of arc is given by following
We’ll find the antiderivative separately
So
Volumes of revolution solids
The volume of a solid which is generated by revolving the region
which is bounded by and the ‐axis on the interval , around
the x-axis, is given by following
78
If the region which is bounded by and the y‐axis on
is revolved around the ‐axis, then its volume is given by fol-
lowing
Example. Find the volume of the solid which is generated by revolv-
ing the region which is bounded by and the ‐axis on ,
around the ‐axis.
Solution:
Figure 17
The volume of the solid is
=
Example. Find the volume of the solid which is generated by revolv-
ing the region which is bounded by and the ‐axis, around
the ‐axis.
79
Solution:
Figure 18
The equation
represents a semicircle with center at
the origin and radius , the solid which is produced by the rotation of the re-
gion by a complete revolution around the x-axis is a sphere. So that the volume of the solid is given by the following
=
Example. Find the volume of the solid which is generated by revolv-
ing the region which is bounded by two curves and ,
around the ‐axis.
Solution:
Figure 19
80
To find the x-coordinates of the intersection’s points of the two curves, we
make , we get
so that
We take an optional value for the interval and let it be .
.
From the above, the volume of the solid is
Example. Find the volume of the solid is generated by revolving
the region which is bounded by the curves of two functions and
, around the ‐axis on the interval
.
Solution:
Figure 20
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