Need Guidance: Area in between Polar Curves

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SUMMARY

The discussion focuses on finding the area of the region that lies inside the circles defined by the polar equations r = 2sin(x) and r = sin(x) + cos(x). The intersection points of these curves occur at x = π/4 and near the origin, which complicates the setup for integration intervals. The area can be calculated using the formula A = (1/2) ∫ from a to b of r^2 dx, where the correct limits of integration must be determined based on the intersection points.

PREREQUISITES
  • Understanding of polar coordinates and polar equations
  • Familiarity with calculus, specifically integration techniques
  • Knowledge of how to find intersection points of polar curves
  • Ability to graph polar equations for visual analysis
NEXT STEPS
  • Learn how to find intersection points of polar curves analytically
  • Study the application of the area formula A = (1/2) ∫ r^2 dx in polar coordinates
  • Explore graphing tools for polar equations to visualize intersections
  • Investigate advanced integration techniques for complex polar regions
USEFUL FOR

Students studying calculus, particularly those focusing on polar coordinates and integration, as well as educators looking for examples of area calculations between polar curves.

GavinMath
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Homework Statement



Find the area of the region that lies inside both of the circles
r = 2sin(x)
r = sin(x) + cos(x)

Homework Equations



A = (1/2)(int from a to b): r^2 dx

(I apologize because I do not know how to make calculus look proper in text form)

The Attempt at a Solution



What I need is some theoretical help. Through graphing these circles I can see that they intersect at pi/4. However, I see that they intersect near the origin, however I can not get a common angle, which makes me confused on how to set up the intervals of my integration. Any ideas to get me going would be much appreciated!
 
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GavinMath said:

Homework Statement



Find the area of the region that lies inside both of the circles
r = 2sin(x)
r = sin(x) + cos(x)

Homework Equations



A = (1/2)(int from a to b): r^2 dx

(I apologize because I do not know how to make calculus look proper in text form)

The Attempt at a Solution



What I need is some theoretical help. Through graphing these circles I can see that they intersect at pi/4. However, I see that they intersect near the origin, however I can not get a common angle, which makes me confused on how to set up the intervals of my integration. Any ideas to get me going would be much appreciated!

What about setting the r values in the two equations equal, and solving for the angle?
 

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