Double Integrals in Polar Coordinates
Introduction
Double integrals in polar coordinates are a powerful tool in calculus that allow us to integrate over regions with circular or radial symmetry. While double integrals in Cartesian (rectangular) coordinates are useful for many applications, certain types of problems are more naturally solved using polar coordinates.
Polar coordinates represent points in the plane using a distance from the origin, r, and an angle, , measured from the positive x-axis. This coordinate system is especially advantageous when working with regions bounded by circles, cardioids, roses, and other curves that can be described more simply in polar form.
The concept extends naturally from single variable calculus where we sometimes use substitution to simplify integrals. In multivariable calculus, changing coordinate systems can dramatically simplify the evaluation of double integrals.
Conversion between Cartesian and Polar Coordinates
To transform between Cartesian and polar coordinates, we use the following relationships:
y = r sin()
r = x + y
tan() = y/x
When changing a double integral from Cartesian to polar coordinates, we must change both the integrand and the area element. In Cartesian coordinates, a small rectangle has area dA = dx dy. In polar coordinates, a small "polar rectangle" bounded by r, r + dr, , and + d has area dA = r dr d.
This additional factor of r appears because the "width" of the region varies with the distance from the origin. Therefore, the general transformation formula is:
Setting Up Double Integrals in Polar Coordinates
Setting up a double integral in polar coordinates requires determining the appropriate limits of integration for r and . The approach involves:
- Sketching the region of integration in the xy-plane.
- Expressing the boundaries of the region in polar coordinates.
- Determining the range of values that sweep through the entire region.
- For each , determining the range of r values that lie within the region.
For common regions, the limits of integration are straightforward:
- For a circle of radius a centered at the origin: 0 r a, 0 2
- For the region between two circles of radii a and b (with a < b) centered at the origin: a r b, 0 2
- For a sector of a circle with angle between and : 0 r a,
For more complex regions, the limits of integration may be functions. For example, for the region inside the circle r = 2a cos(), the limits would be:
Examples of Double Integrals in Polar Coordinates
Example 1: Integral over a Circle
Calculate the area of a circle with radius R using a double integral in polar coordinates.
Solution:
The circle is described by 0 r R, 0 2. The area is:
= [r/2] d
= (R/2) d
= (R/2) d
= (R/2)[]
= (R/2)(2)
= R
Example 2: Integral of a Function
Evaluate the integral e(xy) dA over the entire xy-plane.
Solution:
In polar coordinates, x + y = r, and the entire xy-plane corresponds to 0 r , 0 2. Therefore:
Using the substitution u = r, du = 2r dr, we have r dr = du/2:
= [-e/2]^ d
= (0 + 1/2) d
= (1/2)[]
=
Applications of Double Integrals in Polar Coordinates
Double integrals in polar coordinates have various important applications in mathematics, physics, and engineering:
- Area Calculation: The area of a region D is given by dA = r dr d over the region.
- Volume Calculation: The volume under a surface z = f(x,y) above a region D in the xy-plane is V = f(x,y) dA = f(r cos, r sin) r dr d.
- Center of Mass: For a lamina with density function (x,y), the center of mass coordinates are: x = ( x(x,y) dA) / M
= ( y(x,y) dA) / M - Moment of Inertia: The moment of inertia about different axes can be calculated using: I = y(x,y) dA
I = x(x,y) dA
I = (x + y)(x,y) dA (about the origin) - Average Value: The average value of a function over a region D is: f_avg = (1/Area(D)) f(x,y) dA
Example: Center of Mass of a Disk with Non-Uniform Density
Find the center of mass of a circular disk of radius R if the density at any point is proportional to its distance from the center.
Solution:
The density function is (r) = kr, where k is a constant. In polar coordinates, the region is described by 0 r R, 0 2.
First, calculate the total mass:
= kr dr d
= k[R/3] d
= (kR/3) d
= (2kR)/3
Next, calculate the moments:
= k r sin dr d
= k [R/4] sin d
= kR/4 sin d
= kR/4 [ -cos ]
= kR/4 (-1 + 1)
= 0
Similarly, for the moment about the y-axis:
= k r cos dr d
= k [R/4] cos d
= kR/4 cos d
= kR/4 [ sin ]
= kR/4 (0 - 0)
= 0
Therefore, the coordinates of the center of mass are:
= M/M = 0
Thus, the center of mass is at the origin, which makes sense due to the symmetry of both the region and the density function.
Conclusion
Double integrals in polar coordinates provide a powerful technique for solving problems with circular or radial symmetry. They simplify calculations that would be cumbersome in Cartesian coordinates, especially when working with regions bounded by curves that can be easily expressed in polar form.
The key to working with double integrals in polar coordinates is remembering the extra factor r in the area element, as well as correctly determining the limits of integration. With practice, one can develop the ability to visualize regions in polar coordinates and set up integrals efficiently.
These integrals are essential tools in calculus with wide applications in physics, engineering, and mathematics, particularly in problems involving fields that have radial symmetry, such as gravitational fields, electric fields, and heat conduction problems.
Piper Betle L., N Hexane Fraction, Non Polar, Semi Polar, Polar, GC MS Analysis dan Link D...
Parametric Equations And Polar Coordinates and Reference File Download Link