Conversion of cartesian coordinates to polar coordinates

In summary, the conversation discusses a question about double integration in polar form and concerns about the limits of integration. It is suggested to use a trig substitution and the area of integration is clarified to be separate from the actual integration. The conversation concludes with appreciation for the helpful forum.
  • #1
CostasDBD
2
0
1. Was wondering if anyone could help me confirm the polar limits of integration for the below double integral problem. The question itself is straight forward in cartesian coordinates, but in polar form, I'm a bit suspect of my theta limits after having sketched the it out. any help much appreciated.



2. Homework Equations

[tex]\int^{6}_{0}\int^{y}_{0}xdxdy[/tex]

in polar form:

[tex]\int^{\frac{\pi}{2}}_{\frac{\pi}{4}}\int^{6cosec\theta}_{0} r^{2}cos\theta dr d\theta [/tex]


The Attempt at a Solution



Using a trig substitution over pi/2 and pi/4, i get an answer of 36. it's just when i sketch it, i get a triangle which only has half that area. am i missing something obvious? cheers
 
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  • #2
Welcome to PF!

Hi CostasDBD! Welcome to PF! :smile:

(have a pi: π and a theta: θ and an integral: ∫ :wink:)
CostasDBD said:
Using a trig substitution over pi/2 and pi/4, i get an answer of 36. it's just when i sketch it, i get a triangle which only has half that area. am i missing something obvious? cheers

Yup! :redface:

∫∫ x dxdy isn't the area! :wink:
 
  • #3
Remember, the area of integration and the actual integration are 2 separate entities. Only when the function you're integrating over is f(x,y) = 1 is the integral equal to the area of your area of a double integration.
 
  • #4
Thanks guys. funny what you pick up when you go back a few pages and have a quick read hey.
this forum is fantastic.
 

1. What is the formula for converting cartesian coordinates to polar coordinates?

The formula for converting cartesian coordinates (x, y) to polar coordinates (r, θ) is:
r = √(x^2 + y^2)
θ = arctan(y/x)

2. What are the advantages of using polar coordinates over cartesian coordinates?

Polar coordinates are useful for representing circular and rotational data, making them particularly applicable in disciplines such as physics and engineering. They also simplify certain mathematical operations, such as differentiation and integration.

3. Can negative values be represented in polar coordinates?

Yes, negative values can be represented in polar coordinates. The distance (r) can be negative, indicating a point in the opposite direction from the origin. The angle (θ) can also be negative, indicating a clockwise direction from the reference axis.

4. How do you convert polar coordinates to cartesian coordinates?

To convert polar coordinates (r, θ) to cartesian coordinates (x, y), use the following formulas:
x = r * cos(θ)
y = r * sin(θ)

5. Can you use polar coordinates to represent 3-dimensional space?

No, polar coordinates are limited to representing 2-dimensional space. To represent 3-dimensional space, you would need to use spherical coordinates, which include an additional coordinate for the z-axis.

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