Calculating volume using triple integrals

[kikifast4u]

Homework Statement

Find the volume of the solid enclosed between the cylinder x²+y²=9 and planes z=1 and x+z=5

Homework Equations

V=∫∫∫dz dy dz

The Attempt at a Solution

The problem I have here is setting the integration limits. I first tried using:

• z from 1 to 5-x
• y from √(9-x²) to -3
• x from -3 to 3

However, that gave me a negative answer, so I doubt that's the way to do it.

I then used polar coordinates for x and y and used the integral:
V=∫∫∫r dz dr dθ with limits

• z from 1 to 5-r*cos(θ)
• r from 0 to 3
• θ from 0 to 2∏

This time I got a positive answer, but I'm not sure whether the method is correct. We were never taught to use polar coordinates in volume integrals, so I'm not sure whether it's fine to mix them up.

 

[kikifast4u]

I'm still struggling with this. Any help guys?

 

[HallsofIvy]

Homework Statement

Find the volume of the solid enclosed between the cylinder x²+y²=9 and planes z=1 and x+z=5

Homework Equations

V=∫∫∫dz dy dz

The Attempt at a Solution

The problem I have here is setting the integration limits. I first tried using:

• z from 1 to 5-x

• 
  Okay, that's good.
  
  

  [*]y from √(9-x²) to -3

  Why -3? The cylinder x²+ y²= 9 goes from [itex]-\sqrt{9- x^2}[/itex] to [itex]\sqrt{9- x^2}[/itex].
  

  [*]x from -3 to 3

However, that gave me a negative answer, so I doubt that's the way to do it.

I then used polar coordinates for x and y and used the integral:
V=∫∫∫r dz dr dθ with limits

• z from 1 to 5-r*cos(θ)
• r from 0 to 3
• θ from 0 to 2∏

This time I got a positive answer, but I'm not sure whether the method is correct. We were never taught to use polar coordinates in volume integrals, so I'm not sure whether it's fine to mix them up.

 

[kikifast4u]

Thank you very much! I got the same answer as in the second method, so both are fine. Stupid stupid mistake!