Finding the Volume using a triple integral

[zzz3293]

Homework Statement

Find the volume of the solid bounded by the cylinder x^2+y^2=9 and the planes y+z=5 and z=1

Homework Equations

None

The Attempt at a Solution

My main problem is setting up the integral. So far what I have is 1 as the integrand, my order of integration is dydxdz and my bounds are 0<z<1 and sqrt(9-x^2)<y<5 and I can't figure out x. Is what I have right? What do I need to do? I am really struggling with setting up the bounds for triple integrals

 

[Mark44]

Have you drawn a sketch of the solid? Your typical volume element is dy*dx*dz. What are the bounds for y? I.e., y = ?? to y = ??. Then what are the bounds for x?

 

[Dustinsfl]

For the bounds, I obtained 0≤x≤3, -sqrt(9-x^2)≤y≤sqrt(9-x^2), and 1≤z≤5-y. You might want to double check it though.

 

[Dustinsfl]

I forgot to mention I went from 0-3 because I multiplied the integrals by 2.

 

[zzz3293]

I drew it in the xy plane and am looking at it in grapher, I understand the y bounds now, but x and z are still confusing, should x be -3<x<3? I have absolutely no idea how to go about solving for z...

 

[Mark44]

Yes, assuming that y is as Dustinfl described.

 

[Dustinsfl]

You can use -3 to 3 but since it is a circle you can go from 0 to 3 and multiply by 2 to compensate for -3 to 0. There are 2 planes one is at z=1 and the other is y+z=5. If z=0, then y=5 and if y=0, then z=5, where x can be anything.

 

[zzz3293]

Got it! Now I just need to evaluate which shouldn't be a problem, thanks so much!

 

[Mark44]

Right. In these types of problems the hardest part is figuring out the limits of integration. Also, Dustinfl's suggestion about exploiting the symmetry is a good one that might reduce the chances of making a calculation error.