Triple Integrals in Spherical Coordinates: $z = \sqrt{3x^2+3y^2}$

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I don't know what to integrate, but maybe rewriting your identity as $\rho\cos{\varphi} = \sqrt{3}\sqrt{\rho^2\sin^2(\varphi)} = \sqrt{3}\rho\sin(\varphi)$ your problem becomes easier. I just used $\sin^2(\theta) + \cos^2(\theta) = 1$.

Well, it follows $\tan(\varphi) = 1/\sqrt{3}$ and then $\varphi = \pi/6$. Geometrically, you can see the constant value when you plot the graph:

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Maybe it helps you to integrate whatever you want (the area of the surface, the volume that the cone contains, ...).

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Tyler
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Tyler

Updated on July 22, 2022

Comments

  • Tyler
    Tyler over 1 year

    Hello I am struggling to find the answer to converting:

    $z = \sqrt{3x^2+3y^2}$ to spherical coordinates!

    My answer I keep getting is...

    $\rho\cos(φ) = \sqrt{3}\sqrt{(\rho\cos(\theta)\sin(\varphi))^2+(\rho\sin(\theta)\sin(\varphi))^2}$

    But my submissions are wrong even after I simplify it... any help would be appreciated!

    • polfosol
      polfosol over 6 years
      What is the integral?
    • Feynman
      Feynman over 6 years
      What's the integrand?