Does My Triple Integral for a Cylinder's Volume Bounded by Planes Seem Correct?

[ninjacookies]

I'm supposed to find the volume of the solid bounded by the cylinder x^2+ y^2 =25, the plane x + y + z =8 and the xy plane.

So I decided to use cylindrical coordinates, in which E is bounded by the cylinder r=5, the plane z = 8 - y -z = 8 - r cos(theta) - r sin(theta) , and theta goes from 0 to 2pi


So my integration was setup as follows:

$$\int_{0}^{2pi} \int_{0}^{5} \int_{0}^{8-rcos \theta - rsin \theta} \ r dzdrd\theta$$


and after about a half page of calculations, I ended up with the answer 200pi

Does this answer seem reasonable? I double checked my calculations and they seem all correct...is my integration setup correctly? The reason I'm asking is because the volume of a regular cylinder is

V=pi r^2 h , and in this situation the volume of the cylinder itself would be V=pi (5)^2 (8) = 200pi

so my answer of 200pi kinda confuses me since the cylinder I'm calculating is bounded by a the two planes x+y+z =8 and the xy plane.

Or is this just symmetry and coincidence that they're both equal?

Thanks

 

[dextercioby]

First of all I'm not sure it is a cylinder.You've the equation of a circle.The 2 planes ("xy" and "x+y+z=8") are not parallel.If they were,u may have made the assumption that the base area is 25pi and the height would be 8 and then the volume would be 8*25pi=200pi...

I'm sure that the answer is correct (i've chacked your integration),for symmetry reasons...

Daniel.

 

[ninjacookies]

First of all I'm not sure it is a cylinder.You've the equation of a circle.The 2 planes ("xy" and "x+y+z=8") are not parallel.If they were,u may have made the assumption that the base area is 25pi and the height would be 8 and then the volume would be 8*25pi=200pi...

I'm sure that the answer is correct (i've chacked your integration),for symmetry reasons...

Daniel.

Thanks much Daniel, greatly appreciated. I assumed it was a cylinder because the problem specifically stated it is a cylinder, so I used cylindrical coordinates accordingly since I thought that'd be the best way to go about it. I graphed the whole thing and I understand that the xy plane and x+y+z =8 form a three dimensional pyramid with xy as the base and the second plane with end points (0,0,8) (8,0,0) and (0,8,0) ...and then the cylinder being cut through by the pyramid. So my integration and answer all seem correct? I just want to make sure I didn't make a fatal error in my setup. Thanks again

 

[dextercioby]

It's important that the intersection between the x+y+z=8 plane & the cylinder is symmetrical wrt the plane z=8...

Daniel.

 

[ninjacookies]

It's important that the intersection between the x+y+z=8 plane & the cylinder is symmetrical wrt the plane z=8...

Daniel.

Is there any easy way to determine this? I'm assuming the interesection is symmetrical, since the pyramid has length 8 on all sides and the cylinder is uniform. Though I could be wrong...