How do I solve this multiple integral in the given domain?

[Aner]

Homework Statement

Hi, I have a problem with the following exercise.
Let C={(x,y,z)∈ℝ³ : x²+y²+z²≤1, z≥√(3x²+8y²)} be a subset of ℝ³. Calculate ∫∫∫_C z dxdydz.

Homework Equations

Spherical coordinates, given by x=ρsinΦcosθ, y=ρsinΦsinθ, z=ρcosΦ, and cylindrical coordinates which are x=ρcosα, y=ρsinα, z=z

The Attempt at a Solution

The problem is that, starting from the set C I believe that √(3x²+8y²)≤z≤√(1-x²-y²) so that I should integrate the "∫zdz part" between these two extremes. But when it comes to the conditions on x and y I don't know how to proceed further. I tried to use spherical coordinates so that the first condition becomes ρ²≤1 but then the second one becomes cos(Φ)≥√(3sin²(Φ)cos²(θ)+8sin²(Φ)sin²(θ)) which is worse than the original condition. I also tried to use cylindrical coordinates so that even though the first condition does not improve much the second one becomes z²≥3ρ²cos²(α)+8ρ²sin²(α), which is better than the one I obtained using spherical coordinates, but even so I didn't go very far. Obviously there is some idea I am missing, what is in your opinion the best way to go on?

 

[STEMucator]

You are right about using spherical co-ordinates. Applying the transformation will yield $0 \leq \rho \leq 1$ as you have found already.

The limits for $\theta$ can be found by using your imagination. Are you integrating over a full $2 \pi$ or only half of that? If you sketch the region, it will be clear what the limits for $\theta$ are.

As for $\phi$, you have the right idea, but you seem to have stopped mid-computation. Once you get here:

$$\cos( \phi ) \geq \sqrt{3 \cos^2( \theta ) \sin^2( \phi ) + 8 \sin^2( \theta ) \sin^2( \phi )}$$

Factor out $\sin^2( \phi )$ to obtain:

$$\cot( \phi ) \geq \sqrt{3 \cos^2( \theta ) + 8 \sin^2( \theta )}$$

 

[Ray Vickson]

Homework Statement

Hi, I have a problem with the following exercise.
Let C={(x,y,z)∈ℝ³ : x²+y²+z²≤1, z≥√(3x²+8y²)} be a subset of ℝ³. Calculate ∫∫∫_C z dxdydz.

Homework Equations

Spherical coordinates, given by x=ρsinΦcosθ, y=ρsinΦsinθ, z=ρcosΦ, and cylindrical coordinates which are x=ρcosα, y=ρsinα, z=z

The Attempt at a Solution

The problem is that, starting from the set C I believe that √(3x²+8y²)≤z≤√(1-x²-y²) so that I should integrate the "∫zdz part" between these two extremes. But when it comes to the conditions on x and y I don't know how to proceed further. I tried to use spherical coordinates so that the first condition becomes ρ²≤1 but then the second one becomes cos(Φ)≥√(3sin²(Φ)cos²(θ)+8sin²(Φ)sin²(θ)) which is worse than the original condition. I also tried to use cylindrical coordinates so that even though the first condition does not improve much the second one becomes z²≥3ρ²cos²(α)+8ρ²sin²(α), which is better than the one I obtained using spherical coordinates, but even so I didn't go very far. Obviously there is some idea I am missing, what is in your opinion the best way to go on?

Personally, I would not use cylindrcal or spherical coordinates; I would first just attempt to determine the $(x,y)$ region $R_{xy}$, which is the region where $\sqrt{1-x^2-y^2} \geq \sqrt{3x^2+8y^2}.$
Then I would do the integral
$$\int \! \int_{R_{xy}} \left( \int_{\sqrt{3x^2+8y^2}}^{\sqrt{1-x^2-y^2}} z \, dz \right) \, dx dy .$$

 

[STEMucator]

Personally, I would not use cylindrcal or spherical coordinates; I would first just attempt to determine the $(x,y)$ region $R_{xy}$, which is the region where $\sqrt{1-x^2-y^2} \geq \sqrt{3x^2+8y^2}.$
Then I would do the integral
$$\int \! \int_{R_{xy}} \left( \int_{\sqrt{3x^2+8y^2}}^{\sqrt{1-x^2-y^2}} z \, dz \right) \, dx dy .$$

This way is nice too because you can switch over to polar co-ordinates once you obtain an $xy$ integral.

 

[Aner]

Thank you very much! I will try to use both the approaches. I didn't go further with the calculations of the second conditions using spherical coordinates because I didn't understand how to use that condition but now I understand that I needed your last passage, and I didn't really thought that I could switch to spherical coordinates midway through the integration. I will try to solve the integral and when I'm done I will post my solution here