Double integral to find the area of the region enclosed by the curve

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SUMMARY

The discussion focuses on using a double integral to calculate the area enclosed by the polar curve defined by the equation r = 4 + 3 cos(θ). The minimum value of r is established as 1, leading to the integration limits of 1 ≤ r ≤ 4 + 3 cos(θ). The recommended range for θ is from 0 to π, with the final area being doubled to account for the full region. The curve is identified as a cardioid.

PREREQUISITES
  • Understanding of polar coordinates and curves
  • Knowledge of double integrals in calculus
  • Familiarity with the properties of cardioids
  • Ability to sketch polar graphs
NEXT STEPS
  • Study the process of setting up double integrals for polar coordinates
  • Learn how to sketch and analyze cardioid curves
  • Explore the application of double integrals in calculating areas
  • Investigate the implications of integrating over different ranges in polar coordinates
USEFUL FOR

Students and educators in calculus, mathematicians interested in polar coordinates, and anyone looking to deepen their understanding of double integrals and area calculations in polar systems.

PhysicsMajor
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Greetings all,

I need help setting up this problem:

Use a double integral to find the area of the region enclosed by the curve

r=4+3 cos (theta)

Thanks
 
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Hello.

1)You'll have to draw a polar curve to help you out with this question. From the drawn polar curve, you'll get the minimum value of r to be 1. (when cos(theta) is negative)
2) Thus it follows that the range for r is 1<=r<=4+3cos(theta)
Hence we'll integrate r from 1 to 4+3cos(theta)
3) For the range of theta, you can use the range from 0 to pi for simplicity in calculations.(Just multiply the answer by 2 to obtain the full area.)

I hope this helps =)
 
Yeah,a plot might help u convince why the limit wrt [itex]\theta[/itex] need to be 0 and [itex]\pi[/itex] and why you shouldn't integrate from 0 to [itex]2\pi[/itex]

I think it's a cardioide.


Daniel.
 
Last edited:

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