Double integral and polar coord

In summary, the conversation discusses the evaluation of a double integral over a unit disk in polar coordinates. It is noted that the integral can be easily evaluated using symmetry, and a hint is given for simplifying the calculations.
  • #1
jenc305
16
0
Please help. Thank you.
 

Attachments

  • New Microsoft Word Document.doc
    21 KB · Views: 246
Last edited:
Physics news on Phys.org
  • #2
Alright. The unit disk is

[tex]S = \{(x, \ y) | x^2 + y^2 \leq 1 \}[/tex]

Changing to polar coordinates,

[tex] S = \{ (\rho \cos{\theta}, \ \rho \sin{\theta}) | \rho \leq 1, \ 0 \leq \theta \leq 2\pi \}[/tex]

ie. [tex] x = \rho \cos{\theta}, \ y = \rho \sin{\theta} [/tex].

Thus

[tex] \int \int_S xy\sqrt{x^2 + y^2} dA = \int_0^{2\pi} \int_0^1 \rho \sin{\theta}\rho \cos{\theta} \sqrt{\rho^2 \sin^2{\theta} + \rho^2 \cos^2{\theta}} \ \left|\frac{\partial (x, \ y)}{\partial(\rho, \ \theta)}\right| \ d\rho \ d\theta = \int_0^{2\pi} \int_0^1 \rho^4\cos{\theta}\sin{\theta} \ d\rho \ d\theta[/tex]

I trust you can work out the rest.

I will note that it is quite easy to evaluate this integral without actually doing any calculations, by appealing to symmetry.
 
Last edited:
  • #3
What if D is a closed disk with radius 1 and center (1,0).
That would make the polar coord. x=(r cos(theta)-1) and y=(r sin (theta)).
 
Last edited:
  • #4
It's actually still the same answer. You can make the same symmetry argument.

Here's a hint:

Obviously, if [tex]f[/tex] is [tex]2\pi[/tex]-periodic, then [tex] \int_0^{2\pi} f(x)\sin{x} \ dx = \int_{-\pi}^{\pi} f(x)\sin{x} \ dx[/tex]. Thus if [tex]f[/tex] is an even function, since [tex]\sin[/tex] is odd, then [tex]f(x)\sin{x}[/tex] is odd and

[tex]\int_0^{2\pi} f(x)\sin{x} \ dx = 0[/tex]
 
Last edited:

1. What is a double integral?

A double integral is a type of integral used in multivariable calculus to calculate the volume under a surface in two-dimensional space. It involves taking the limit of a sum of infinitely small rectangles to approximate the total area.

2. How is a double integral related to polar coordinates?

In polar coordinates, a point is described by its distance from the origin (radius) and its angle from a reference line (theta). A double integral in polar coordinates is used to calculate the area under a polar curve by integrating the function with respect to both radius and theta.

3. What is the difference between a single and double integral?

A single integral calculates the area under a curve in one-dimensional space, while a double integral calculates the volume under a surface in two-dimensional space. A single integral has one variable of integration, while a double integral has two variables of integration.

4. What is the benefit of using polar coordinates in a double integral?

Polar coordinates can simplify the calculation of a double integral in cases where the region of integration is more naturally expressed in terms of radius and theta. This makes it easier to set up the integral and can often lead to simpler calculations.

5. How is a double integral used in real-world applications?

Double integrals are used in many fields of science and engineering to calculate volumes, areas, and averages. For example, they can be used in physics to determine the mass of an object with varying density, in economics to calculate the total revenue of a business, or in biology to calculate the average rate of growth of a population.

Similar threads

Replies
1
Views
834
Replies
1
Views
793
  • Calculus
Replies
11
Views
2K
  • Calculus
Replies
24
Views
3K
  • Calculus
Replies
4
Views
2K
  • Calculus
Replies
2
Views
885
  • Calculus
Replies
1
Views
2K
Replies
4
Views
2K
  • Calculus
Replies
1
Views
1K
Replies
2
Views
2K
Back
Top