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Find appropriate parametrization to find area bounded by a curve

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  • #1
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Problem:

Use an appropraite parametrization [tex]x=f(r,\theta), y=g(r,\theta)[/tex] and the corresponding Jacobian such that [tex] dx \ dy \ =|J| dr \ d\theta[/tex] to find the area bounded by the curve [tex]x^{2/5}+y^{2/5}=a^{2/5}[/tex]

Attempt at a Solution:

I'm not really sure how to find the parametrization. Once I have that, calculating the Jacobian is simple. Then all that's left is computing a double integral. Right?
 

Answers and Replies

  • #2
Zondrina
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Problem:

Use an appropraite parametrization [tex]x=f(r,\theta), y=g(r,\theta)[/tex] and the corresponding Jacobian such that [tex] dx \ dy \ =|J| dr \ d\theta[/tex] to find the area bounded by the curve [tex]x^{2/5}+y^{2/5}=a^{2/5}[/tex]

Attempt at a Solution:

I'm not really sure how to find the parametrization. Once I have that, calculating the Jacobian is simple. Then all that's left is computing a double integral. Right?
Hint: Your parametrization hints at polar co-ordinates.

##x = rcos\theta, y = rsin\theta##
 
  • #3
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But if I use ##x=rcos\theta## and ##y=rsin\theta##, then I have ##dx=rd\theta## and ##dy=rd\theta##. Also ##|J|=r##. However we want ##dx \ dy=|J|dr \ d\theta##.

Using ##x=rcos\theta## and ##y=rsin\theta##, then the curve is given by [tex](rcos\theta)^{\frac{2}{5}}+(rsin\theta)^{\frac{2}{5}}=a^{\frac{2}{5}}[/tex]

So to find the area bounded by this curve we want to perform a double integral. I'm just confused on setting it up. Now that we're in terms of ##r## and ##\theta##, how do we figure out the limits of integration for the given curve?
 
Last edited:

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