Finding the Volume using a triple integral

  • Thread starter zzz3293
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  • #1
zzz3293
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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


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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
 

Answers and Replies

  • #2
36,661
8,659
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?
 
  • #3
Dustinsfl
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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.
 
  • #4
Dustinsfl
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I forgot to mention I went from 0-3 because I multiplied the integrals by 2.
 
  • #5
zzz3293
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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...
 
  • #6
36,661
8,659
Yes, assuming that y is as Dustinfl described.
 
  • #7
Dustinsfl
2,285
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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.
 
  • #8
zzz3293
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Got it! Now I just need to evaluate which shouldn't be a problem, thanks so much!
 
  • #9
36,661
8,659
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.
 

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