Example 2
Evaluate the integral
, where the region of integration R lies between the circles
and
.
Solution.
In polar coordinates, the region of integration R is the polar rectangle (Figure 5):
Fig.5
So using the formula
we find the integral:
Example 5
Calculate the double integral
by transforming to polar coordinates. The region R is the disk
Solution.
The region R is presented in Figure 9.
Fig.9
The image S of the initial region R is defined by the set
. The double integral in polar coordinates becomes
We compute this integral using integration by parts:
Let
. Then
. Hence,
Example 1
Calculate the double integrals
. The region of integration R is bounded by
.
Solution.
We can represent the region R in the form
. So R is the region of type I (see Figure 1). According to the Fubini's formula,
Calculate first the inner integral:
Now we can compute the outer integral:
Fig.1
Example 2
Calculate the integral
. The region of integration R is bounded by the lines
.
Solution.
We can represent the region R as the set
The region R belongs to type I. Transforming the double integral into the iterated one, we get
Example 6
Fig.6
Find the integral
, where R is bounded by the line
and parabola
.
Solution.
The region R is shown above in Figure 6. Find the points of intersection of the line and parabola:
Hence, the given lines bounded the region R intersect at the points (−3,−6) and (1,2). Then by the Fubini's theorem,
the integral is
Example 7
Calculate the double integral
, where R is bounded by
.
Solution.
The region of integration R is defined by the set
Applying the Fubini's theorem, we obtain
and shown below in Figure 7.
To find the latter integral, we make the substitution:
When x = 0, we have z = 0. Hence at x = 1 we have z = 1. Then the integral is
Fig.7
Example 2
Find the area of the region R bounded by
Solution.
.
We first determine the points of intersection of the two curves.
So the coordinates of the points of intersection are
The given region R is shown in Figure 5 below. It is simpler to consider R as a type II region. To calculate the area of
the region, we transform the equations of the boundaries:
Then we have
Fig.5
Example 9
Use polar coordinates to find the volume of a right circular cone with height H and a circular base with radius R (see
Figure 15).
Solution.
Fig.15
Fig.16
We first find the equation of the cone. Using similar triangles (Figure 16), we can write
Hence,
Then the volume of the cone is
Example 3
Electric charge is distributed over the disk
Calculate the total charge of the disk.
so that its charge density is
.
Solution.
In polar coordinates, the region occupied by the disk is defined by the set
charge is
. The total