Integral homework problem help

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  • #1
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[tex]2c\int_{x=-a}^a\int_{y=-b\sqrt{1-\frac{x^2}{a^2}}}^{b\sqrt{1-\frac{x^2}{a^2}}}\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}dydx[/tex]
Can you help me with this integral?
 

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  • #2
arildno
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You have already been advised to do a change of variables, rather than do this in Cartesian variables.
 
  • #4
arildno
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Well, how would you go about proving that the unit ball has volume [itex]\frac{4}{3}\pi[/itex] ?
 
  • #5
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Well, the idea is probably good, but it doesn't help me with the integral
 
  • #6
HallsofIvy
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Well, the idea is just to use spherical coordinates. Have you sketched the region over which that integral is taken? Looks to me like there is a heckuvalot of symmetry there!
 
  • #7
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Won't it be much more complicated, or is it the only way?
r=(x^2+y^2+z^2)^1/2.
 
Last edited:
  • #8
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It's clear that there is symmtry, but if r=(x^2+y^2+z^2)^1/2 everything will be much more complicated. How should it be solved then?
 
  • #9
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Sorry for this post, I had some problems with my internet browser.
 
  • #10
siddharth
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-=nobody=-, as everyone has already said on this thread, transform your coordinates. ie, set

[tex] x= a r \cos\theta [/tex]

[tex] y = b r\sin \theta [/tex]

Now, find the Jacobian and limits of integration of [itex] \theta [/itex] and [itex] r [/itex]. Can you take it from here?
 
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