Solving Green's Theorem Integrals: Finding the Area

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SUMMARY

This discussion focuses on applying Green's Theorem to solve integrals related to area calculations. The user demonstrates the transformation of parametric equations, specifically using x = sin(t)cos(t) and y = sin²(t), leading to the Cartesian equation x² + y² - y = 0. By completing the square, the user identifies the resulting shape as a circle, which is crucial for applying Green's Theorem effectively. The solution emphasizes the importance of converting parametric forms to Cartesian coordinates for area determination.

PREREQUISITES
  • Understanding of Green's Theorem in vector calculus
  • Familiarity with parametric equations and their Cartesian equivalents
  • Knowledge of completing the square in algebra
  • Basic proficiency in integral calculus
NEXT STEPS
  • Study the applications of Green's Theorem in various integral problems
  • Learn about converting parametric equations to Cartesian coordinates
  • Explore examples of area calculations using Green's Theorem
  • Investigate the implications of different shapes on integral evaluations
USEFUL FOR

Students in calculus courses, particularly those studying vector calculus, as well as educators looking for practical examples of Green's Theorem applications.

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Homework Statement



Hey.
I need to use Green's theorem in order to solve this integral.
My question is, how can I find the area for the Green's theorem integral?

Homework Equations





The Attempt at a Solution

 

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If x= sin t cos t and y= sin2 t, then x2+ y2= sin2tcos2t+ sin4t= sin2 t(cos2 t+ sin2t)= sin2(t)= y so the equation, in Cartesian coordinates, is x2+ y2- y= 0. Complete the square and you can see that is a circle.
 
Last edited by a moderator:
HallsofIvy said:
If x= sin t cos t and y= sin2 t, then x2+ y2= sin2cos2+ sin[sup4[/sup]= sin2 t(cos2 t+ sin2t)= sin2(t)= y so the equation, in Cartesian coordinates, is x2+ y2- y= 0. Complete the square and you can see that is a circle.

thanks.
 

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