Double Integrals in Polar Form

University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
12
Double Integrals in Polar Form
Integrals are sometimes easier to evaluate if we change to polar coordinates. This section shows how to accomplish the change and how to evaluate integrals over regions whose boundaries are given by polar equations.
Integrals in Polar Coordinates
? Suppose that a function ƒ(r, ?) is defined over a region R that is bounded by the rays ? = ?? and ? = ?? and by the continuous curves r = g1(?) and r = g2(?).
? Suppose also that 0 ? g1(?) ? g2(?) ? a for every value of ? between ?? and ?? Then R lies in a fan-shaped region Q defined by the inequalities 0 ? r ? a and ?? ? ? ? ??.
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
13
Finding Limits of Integration
EXAMPLE: Find the limits of integration ƒ(r, ?) for integrating over the region R that
lies inside the circle x2+y2= 4 and outside the y = ? .
Solution:
1. Sketch: Sketch the region and label the bounding curves.
2. Find the r-limits of integration: Imagine a ray L from the origin cutting through R in the direction of increasing r. Mark the r-values where L enters and leaves R. These are the r-limits of integration. They usually depend on the ? angle that L makes with the positive x-axis.
3. Find the ?-limits of integration: Find the smallest and largest ? -values hat bound R. These are the u-limits ? -limits
The integral is
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
14
Area in Polar Coordinates
The area of a closed and bounded region R in the polar coordinate plane is
EXAMPLE 7: Find the limits of integration ƒ(r, ?) for integrating over the region R
that lies inside the cardioid r = 1 + cos? and outside the circle r = 1.
Solution:
1. We first sketch the region and label the bounding curves.
2. Next we find the r-limits of integration. A typical ray
from the origin enters R where r = 1 and leaves where
r = 1 + cos ?.
3. Finally we find the u-limits of integration. The rays from the origin that intersect R
run from ? = - ???2 to ? = ???2 the integral is
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
15
EXAMPLE 8: Find the area enclosed by the lemniscate r 2 = 4 cos2?.
Solution: Graph the lemniscate to determine the limits of integration and see from
the symmetry of the region that the total area is 4 times the first-quadrant
portion.
Changing Cartesian Integrals into Polar Integrals
The procedure for changing a Cartesian integral b y i n t o a
polar integral has two steps. First substitute x = r cos ? and y
= r sin ? and replace dx dy by r dr d? in the Cartesian integral. Then supply polar
limits of integration for the boundary of R. The Cartesian integral then becomes
where G denotes R
The region of integration in polar coordinates.
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
16
EXAMPLE 9: Find the polar moment of inertia about the origin of a thin plate of
density ?(x, y) = 1 bounded by the quarter circle x2 + y2 = 1 in the
first quadrant.
Solution: We sketch the plate to determine the limits of integration . In Cartesian
coordinates, the polar moment is the value of the integral
an integral difficult to evaluate without tables. Things go better if we change the original
integral to polar coordinates. Substituting
EXAMPLE 10: Evaluating integrals using Polar coordinates
where R is the semicircular region bounded by the x-axis
and the curve y = ?
Solution: