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# Integrate this challenging question Watch

1. Hey,

I'm stuck with this question:
Calculate the volume enclosed by a cylinder with equation and the planes and

Can anybody give me some direction?
Thank you
2. Should be able to do it from basic formulas and a bit of symmetry.
3. For your 3rd boundary condition do you mean z=5 rather than x+z=5 ?
4. (Original post by ve9jonny)
For your 3rd boundary condition do you mean z=5 rather than x+z=5 ?
I wish would make things a lot easier wouldn't it?
5. If yes then (pi x r^2 x h) = pi x 9 x 5 = 45pi which is simple and involves no integration.

If no then it is not a cylinder as far I can tell
6. (Original post by ve9jonny)
If yes then (pi x r^2 x h) = pi x 9 x 5 = 45pi which is simple and involves no integration.

If no then it is not a cylinder as far I can tell
No, it's not a cylinder.
7. The second plane slices the cylinder at an angle... first thing to do is work out the highest and lowest points of this slice .

You'll find these points by using the line x + z = 5 along the diameter of the cylinder, y=0. You are able to choose the value since the plane is independent of y. So solve simultaneously for that line and the cylinder at y=0).

Once you have the heights (z-coordinates) of these points, the volume is simply the cylinder to the lower point plus half the remaining portion (since by looking in the y direction it is clear you have half this last bit of cylinder: one triangle from a rectangle).

Hope this makes things clearer.
8. (Original post by Visa Electron)
No, it's not a cylinder.
The volume bellow the z=f(x,y) and above the area of A on the z=0 plane
and

Your A is the base of the cylinder that is
So

But it is better to calculate with polar coordinates
and

and limits of 0 and

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