Finding the area of one loop of the lemniscate

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Homework Help Overview

The problem involves finding the area of one loop of the lemniscate defined by the equation r² = a²sin(2θ) where a > 0, using double integration techniques.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the limits of integration for both θ and r, with one participant noting they have determined the limits for θ but are uncertain about the limits for r. Others suggest that the limits for r depend on θ and should be functions of θ, prompting questions about how to set these limits correctly.

Discussion Status

The discussion is ongoing, with participants seeking clarification on how to establish the limits of integration for r in relation to θ. There is a focus on understanding the integration process and the relationship between the variables involved.

Contextual Notes

Participants express confusion regarding the integration process, particularly in determining the appropriate limits for r based on the curve defined by the lemniscate equation.

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Homework Statement



Find the area of the region bounded by one loop of the lemniscate r2 = a2sin(2θ) with a > 0 using double integration.


Homework Equations





The Attempt at a Solution



I was able to figure out the limits of integration for theta (0 to ∏/2), but what would my limits be for r?
 
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The limits for r will be functions of theta. Integrate wrt r first. When you plug in the limits, those functions will come into the integrand. Then you can integrate wrt theta.
 
I'm sorry, I really don't understand. How can I integrate wrt r without setting the limits of my integrand first? What would those limits be?
 
annpaulveal said:
I'm sorry, I really don't understand. How can I integrate wrt r without setting the limits of my integrand first? What would those limits be?

When you integrate in the r direction, r goes from r = 0 to the r on the curve, which is a function of ##\theta##.
 
annpaulveal said:
I'm sorry, I really don't understand. How can I integrate wrt r without setting the limits of my integrand first? What would those limits be?
For a given value of theta, what is the smallest value of r within the region? What is the largest value with the region?
 

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