(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use polar coordinates to find the volume bounded by the paraboloids z=3x^{2}+3y^{2}and z=4-x^{2}-y^{2}

2. Relevant equations

3. The attempt at a solution

Somehow, through random guessing, I managed to get the right answer, it's just that I don't understand how I got it. Also, because the z is involved, I actually used cylindrical coordinates, but would that still be considered the same thing as polar coordinates? So anyway, I changed the two paraboloid equations to z=3r^{2}and z=4-r^{2}. Then setting these two equations equal to each other (since they are both equal to z), I solved for r and got the limits of -1,1. For the limits of theta, I just happened to take it from 0 to 2pi. Lastly for the z limits, I just tried from 4-r^{2}to 3r^{2}, so that gave me the equation:

[tex]\int^{2\pi}_0\int^1_{-1}\int^{3r^2}_{4-r^2}dzrdrd\theta[/tex]

However, solving this equation didn't give me the right answer, so I changed the limits of r to 0 to 1, and switched the z-limits around, so now it is:

[tex]\int^{2\pi}_0\int^1_0\int^{4-r^2}_{3r^2}dzrdrd\theta[/tex]

And solving for this, gave me the right answer of 2pi. The problem is that I don't understand the real logic behind what I did.

So in summary, what I didn't understand was how to establish the limits for theta, r, and z.

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# Volume Using Polar Coordinates

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