Limits of integration on Polar curves

In summary, the conversation discusses determining the limits of integration for a polar curve, which is a closed curve defined in polar coordinates such as cardioids, limacons, and circles. The speaker expresses difficulty in finding a clear explanation online and prefers using graph trace to gain an intuitive understanding. They also mention that usually the limits of integration for theta run from 0 to 2pi or from -pi to pi, but this may not be helpful for everyone.
  • #1
CrazyNeutrino
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General question, how do you determine the limits of integration of a polar curve? Always found this somewhat confusing and can't seem to find a decent explanation on the internet.
 
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  • #2
What's a polar curve ? A circle on the South pole ? A trajectory described in polar coordinates ?
Please give a clearer description.
 
  • #3
Oh sorry, any closed curve defined in Polar coordinates. Cardiods, limacons, circles, the works.
 
  • #4
I should have asked straight away too: What is it you want to integrate ? some function over the surface, over the boundary ? A vector function ? Just the circumference or the area ?
 
  • #5
The area enclosed by the Polar curve using Int(1/2 r^2) d theta. I find the determination of the limits of integration slightly ambiguous when I watch any tutorials or read up on Polar coordinates. I normally just use graph trace but I'd like to get an intuitive understanding
 
  • #6
Browsing some of the links at the lower left might be instructive.
We did a cardioid here not so long ago (no full solution, just hints).
Point is: with a concrete example we can see where things go wrong for you.
As you can see in the link, I am an advocate of your approach:
CrazyNeutrino said:
normally just use graph trace
and from that, with experience, grows intuition. The latter two aren't sold by weight (in contrast with what some managers seem to think).

I don't think I can provide much guidance based on e.g.
CrazyNeutrino said:
I find the determination of the limits of integration slightly ambiguous
All I can say is usually ##\theta## runs from 0 to ##2\pi## or from ##-\pi## to ##+\pi##. But I doubt if that is helpful for you.
 
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Likes CrazyNeutrino

1. What are the limits of integration on polar curves?

The limits of integration on polar curves are typically determined by the range of values for the variable θ. This can be determined by looking at the graph of the polar curve and identifying the starting and ending points for θ.

2. How do I find the limits of integration for a specific polar curve?

To find the limits of integration for a specific polar curve, you can use the equation for the curve and plug in values for θ to determine the range of values that make up the curve. Alternatively, you can also look at a graph of the curve to visually determine the limits of integration.

3. Can the limits of integration for a polar curve be negative?

Yes, the limits of integration for a polar curve can be negative. This typically occurs when the curve has a portion that extends below the x-axis, resulting in negative values for θ. It is important to consider both positive and negative values when determining the limits of integration for a polar curve.

4. How do the limits of integration affect the area under a polar curve?

The limits of integration determine the range of values for which the area under the polar curve is calculated. Changing the limits of integration can result in a different area value, as it includes a different portion of the curve. It is important to ensure that the limits of integration accurately capture the desired area under the curve.

5. Can the limits of integration for a polar curve change during the integration process?

Yes, the limits of integration for a polar curve can change during the integration process. This can occur when the curve has multiple regions with different values for θ. In these cases, the limits of integration will need to be adjusted to accurately capture the entire area under the curve.

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