# Need help with the boundaries of a 3D solid

• Feb 13th 2010, 10:03 AM
qwesl
Need help with the boundaries of a 3D solid
How do you find the boundaries of a volume defined by
$\displaystyle -1\leq x+y+3z\leq 1$
$\displaystyle 1\leq 2y-z\leq 7$
$\displaystyle -1\leq x+y\leq 1$

to compute $\displaystyle \int\int\int 9z^2 dxdydz$
• Feb 13th 2010, 10:57 AM
TKHunny
Various ways. Projection is one such way.

Simply eliminate all the 'z's and see wht it looks like from the point of view of the x-y plane.

Do the same for x-z and y-z.

Are you sure you must use dxdydz? Often there is one way that is easier than another. Maybe dydzdx would be more straight-forward. You must keep your eyes open.

Exploit symmetries and redundancies. I don't see much in this one except that the first and third constraints are a little odd. If z > 0, does the third one mean anything? Subtract 3z in the first and compare it to the third. What does that mean?
• Feb 13th 2010, 12:46 PM
qwesl
Quote:

Originally Posted by TKHunny
Various ways. Projection is one such way.

Simply eliminate all the 'z's and see wht it looks like from the point of view of the x-y plane.

Do the same for x-z and y-z.

Are you sure you must use dxdydz? Often there is one way that is easier than another. Maybe dydzdx would be more straight-forward. You must keep your eyes open.

Exploit symmetries and redundancies. I don't see much in this one except that the first and third constraints are a little odd. If z > 0, does the third one mean anything? Subtract 3z in the first and compare it to the third. What does that mean?

I used the constraints to my advantage and this is what I came up with. Can anybody confirm whether the following is correct?

$\displaystyle \int_{\frac{-15}{6}}^{\frac{5}{6}} \int_{\frac{3+2x}{2}}^{\frac{5+2x}{2}}\int_{\frac{-1-2x-2z}{4}}^{\frac{9-2x-2z}{4}} 9z^2 dydzdx$