Finding the Area Bounded by a Polar Curve: A Proof of Integrability Criteria

In summary, the conversation is about determining the expression for the area bounded by a polar curve and discussing the criteria for integrability using both Darboux and Riemann sums. The person is seeking feedback on their proof and has found a better method to convert the file to a pdf. They also mention that their attachments are pending approval.
  • #1
jgens
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Homework Statement



Determine the expression for the area bounded by a polar curve and the criterion for integrability using both Darboux and Riemann sums.

Homework Equations



N/A

The Attempt at a Solution



Any suggestions on how to correct any errors in the following proof, particularly in the steps determining the criterion for Riemann integrability are much appreciated. I'm not particularly great at proofs so constructive criticism is welcome. I had to convert the file to a pdf and it screwed up a couple of the equations, most notably, a factor of (ti - ti-1) appears in the denominator when it should not. Thanks!
 

Attachments

  • Polar Area _Math_.pdf
    134.5 KB · Views: 258
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  • #2
I found a better method to convert the file to a pdf. Hopefully this will make it easier to read and correct some of the conversion errors from the other pdf.
 

Attachments

  • Polar Area (Math).pdf
    131.6 KB · Views: 247
  • #3
No one has any comments?
 
  • #4
Just so you know that you're not waiting because no one wants to help you, your attachments are

'Attachments Pending Approval'
 

1. What is the equation for finding the area bounded by a polar curve?

The equation for finding the area bounded by a polar curve is A = ½∫θ1θ2 r2(θ) dθ, where r(θ) represents the polar equation and θ1 and θ2 are the angles where the curve intersects.

2. How do you determine the limits of integration for finding the area bounded by a polar curve?

The limits of integration for finding the area bounded by a polar curve are determined by finding the values of θ where the curve intersects. These values are then used as the upper and lower limits of the integral.

3. Can you find the area bounded by a polar curve without using calculus?

Yes, it is possible to find the area bounded by a polar curve without using calculus by approximating the curve with straight lines and calculating the area of each sector. However, this method may not be as accurate as using calculus.

4. What is the difference between finding the area bounded by a polar curve and finding the area under a polar curve?

The area bounded by a polar curve refers to the area enclosed by the curve, while the area under a polar curve refers to the area between the curve and the origin. The former involves finding the area using integration, while the latter can be found by using the formula A = ½r2θ.

5. Can you use the same method to find the area of any polar curve?

Yes, the same method can be used to find the area of any polar curve. However, the limits of integration and the equation may vary depending on the specific polar curve.

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