Triple Integration

TheSetMiner

I'm having some trouble approaching the following triple integration question:
I understand I need to find suitable limits for my triple integral but I'm unsure on how to go about finding these. The examples I've found online have been for much simpler triple integrals.

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Use triple integration to find the volume of the region enclosed by z = x^2 +y^2 and z = 18 − x^2 − y^2.

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Any help is greatly appreciated!

TheBody

TheSetMiner wrote: »

I'm having some trouble approaching the following triple integration question:
I understand I need to find suitable limits for my triple integral but I'm unsure on how to go about finding these. The examples I've found online have been for much simpler triple integrals.

---

Use triple integration to find the volume of the region enclosed by z = x^2 +y^2 and z = 18 − x^2 − y^2.

---

Any help is greatly appreciated!

Hi there! The charter forbids me from doing the problem for you.

To get you started, set the two curves equal to each other to see what you get.

Also think about a change of coordinate system.

Post back if you need more help!

TheSetMiner

TheBody wrote: »

Hi there! The charter forbids me from doing the problem for you.

To get you started, set the two curves equal to each other to see what you get.

Also think about a change of coordinate system.

Post back if you need more help!

Hi, so setting them equal I get x^2 + y^2 = 9, ie a circle at origin of radius 3.

I guess you are referring to polar coordinates here so by changing to this system then I could integrate over 0 to 2pi for z and over limits for x and y from this circle equation?

TheBody

TheSetMiner wrote: »

Hi, so setting them equal I get x^2 + y^2 = 9, ie a circle at origin of radius 3.

I guess you are referring to polar coordinates here so by changing to this system then I could integrate over 0 to 2pi for z and over limits for x and y from this circle equation?

Not sure I understand the bit in bold but I think you have the idea.


Anyway, switching to cylindrical coordinates and noting that [latex]x=r\cos (\theta)[/latex] and [latex]x=r\sin (\theta)[/latex], the z limits will be:

[latex]z=x^2+y^2=r^2[/latex] and
[latex]z=18-x^2-y^2=18-(x^2+y^2)=18-r^2[/latex].

The r limits will be from 0-3 and the [latex]\theta[/latex] limits will run from [latex]0-2\pi[/latex].

TheSetMiner

TheBody wrote: »

Not sure I understand the bit in bold but I think you have the idea.


Anyway, switching to cylindrical coordinates and noting that [latex]x=r\cos (\theta)[/latex] and [latex]x=r\sin (\theta)[/latex], the z limits will be:

[latex]z=x^2+y^2=r^2[/latex] and
[latex]z=18-x^2-y^2=18-(x^2+y^2)=18-r^2[/latex].

The r limits will be from 0-3 and the [latex]\theta[/latex] limits will run from [latex]0-2\pi[/latex].

thanks for the help, so is this the correct integral?:

int[2pi, 0]int[3, 0]int[r^2, 18 - r^2] dz dr dtheta or have I gone wrong?

TheBody

TheSetMiner wrote: »

thanks for the help, so is this the correct integral?:

int[2pi, 0]int[3, 0]int[r^2, 18 - r^2] dz dr dtheta or have I gone wrong?

Almost, you forgot the Jacobian because you switched to cylindarical coords. i.e. you pick up an r in the integration.

[latex]\int_0^{2\pi}\int_0^3\int_{r^2}^{18-r^2} r dz dr d\theta[/latex]

TheSetMiner

ah yeah thanks very much for your help!

TheBody

TheSetMiner wrote: »

ah yeah thanks very much for your help!

No problem. Glad I could help.