Finding Limits for Polar Coordinate Area Integration

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In summary, the conversation is about finding the area bounded by a curve and a half-line in polar coordinates. The method is to set the polar equation equal to zero to find the values of theta where the curve crosses the half-line. Then, the area can be found by using integration with the limits set as the theta values found previously. There were some errors in the calculations, but the overall method was correct. The conversation also briefly touches on the importance of sketching the graph and using the correct coefficients in the equation.
  • #1
daster
I need help with finding areas. I'm having trouble picking the correct limits for my integration.

Say for example we had r=2cos2t and a half-line t=pi/6, and I want to find the small area bounded between them.

[tex]\frac{1}{2}\int (2\cos (2\theta))^{2}\,d\theta[/tex]

I can do the integration, but what do I choose as its limits?

I'm not particularly interested in the answer to this question; I'm looking for a good explanation or maybe a couple of pointers.
 
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  • #2
At what value(s) of theta does [itex]r[/itex] become zero ? Set the r(theta) equation to zero and solve for theta. Use the 1st quadrant value of theta obtained as the upper bound with [itex]\frac{\pi}{6}[/itex] as the lower.
 
  • #3
Why do we want the value of theta at r=0?
 
  • #4
daster said:
Why do we want the value of theta at r=0?

Have you even sketched the graph yet ? It will become clear as day if you do.
 
  • #5
Can you please check my working?

(a) Sketch the curve with polar equation

[tex]r=3\cos 2\theta, \, -\frac{\pi}{4}\leq\theta<\frac{\pi}{4}[/tex].

The curve looks like 1 rose petal.

(b) Find the area of the smaller finite region enclosed between the curve and the half-line [itex]\theta=\frac{\pi}{6}[/itex].

[tex]Area = \frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} (3\cos 2\theta)^2\,d\theta = \frac{9}{4}\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} (1+\cos 4\theta)\,d\theta = \frac{9}{4}\left[\theta+\frac{1}{4}\sin 4\theta\right]_{\frac{\pi}{6}}^{\frac{\pi}{4}} = \frac{3\pi}{16}- \frac{9\sqrt{3}}{32}[/tex]

A couple of arithmetic slips probably made it through, since I did all of this using my keyboard & notepad.exe. :tongue2: So I'm just wondering if my method was correct.
 
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  • #6
The coefficient is 3 ? Not 2 ? If that's the case then your working in the last post is correct. But why did you give a different equation in your first post ? :confused:
 
  • #7
In my first post I made up an equation and a line, and when I went to do a question from my textbook, it happened to be almost like the one I made up. :tongue2:

Anyway, thanks for your help. I appreciate it. :smile:
 
  • #8
Sure. :smile:
 

1. What is polar coordinate area integration?

Polar coordinate area integration is a mathematical technique used to find the area under a curve in polar coordinates. It involves breaking down the curve into smaller sections and approximating the area of each section using polar coordinates, then adding up all the approximations to get the total area.

2. Why do we need to find limits for polar coordinate area integration?

Limit finding is necessary in polar coordinate area integration because the area under a curve in polar coordinates cannot be calculated using a simple formula like in Cartesian coordinates. Therefore, we need to determine the limits of integration in order to accurately calculate the area.

3. What are the methods for finding limits in polar coordinate area integration?

There are two main methods for finding limits in polar coordinate area integration: the vertical limit method and the horizontal limit method. The vertical limit method involves finding the maximum and minimum values of the curve in terms of the angle θ, while the horizontal limit method involves finding the maximum and minimum values of the curve in terms of the radius r.

4. How do we know if we have found the correct limits?

The correct limits for polar coordinate area integration can be determined by graphing the curve in polar coordinates and identifying the region of interest. The limits should encompass the entire region of interest and not include any unnecessary areas outside of it.

5. Can we use polar coordinate area integration for any type of curve?

Yes, polar coordinate area integration can be used for any type of curve as long as it can be expressed in polar coordinates. This includes curves such as circles, cardioids, and roses. However, the method for finding limits may vary depending on the type of curve.

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