Volume Using Polar Coordinates

[squeeky]

Homework Statement

Use polar coordinates to find the volume bounded by the paraboloids z=3x²+3y² and z=4-x²-y²

Homework Equations

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=3r² and z=4-r². 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-r² to 3r², so that gave me the equation:
$$\int^{2\pi}_0\int^1_{-1}\int^{3r^2}_{4-r^2}dzrdrd\theta$$
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:
$$\int^{2\pi}_0\int^1_0\int^{4-r^2}_{3r^2}dzrdrd\theta$$
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.

 

[HallsofIvy]

Homework Statement

Use polar coordinates to find the volume bounded by the paraboloids z=3x²+3y² and z=4-x²-y²

Those two paraboloids intersect when z= 3x²+ 3y²= 4- x²- y² or 4x²+ 4y²= 4 which reduces to x²+ y²= 1. That projects onto the xy-plane as the unit circle. It shouldn't be hard to see that the volume you want is above that circle. To cover that circle, $\theta$ goes from 0 to $2\pi$ and r goes from 0 to 1. Those are your limits of integration.

For each (x, y), the upper boundary is z= 4- x²- y² and the lower boundary is z= 3x²+ 3y². The volume of a "thin" rectangular solid used to construct the Riemann sums for this volume would be [(4- x²- y²)- (3x²+ 3y²)]dxdy= (4- 4x²- 4y²)dxdy= 4(1- x²- y²)dxdy and that, in polar coordinates, is 4(1- r²)rdrd$\theta$.

Homework Equations

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?

Yes, "cylindrical coordinates" is just polar coordinates in the xy-plane.

So anyway, I changed the two paraboloid equations to z=3r² and z=4-r². 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.

No, you didn't. Or you shouldn't. r cannot be negative. r must be between 0 and 1.

For the limits of theta, I just happened to take it from 0 to 2pi.

Did you think at all about the geometry of the situation? This has circular symmetry. To cover the entire volume you have to cover the entire circle: 0 to $2\pi$.

Lastly for the z limits, I just tried from 4-r² to 3r²

Surely you recognized that z= 4- r² is ABOVE z= 3r² for r<= 1?

, so that gave me the equation:
$$\int^{2\pi}_0\int^1_{-1}\int^{3r^2}_{4-r^2}dzrdrd\theta$$
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:
$$\int^{2\pi}_0\int^1_0\int^{4-r^2}_{3r^2}dzrdrd\theta$$
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.