Maximize value that triple integral will compute

[Lavoisier44]

Homework Statement

This is a question an exam I already took but I felt as though the solution posted by the professor might be incorrect so I wanted to hear some other opinions. The question is as follows:
What is the maximum value that \iiint (-|x|-|y|+1)\,dV will compute for any region D \subset R3

We were supposed to come up with the bounds for (so I thought) the triple integral and then integrate the given integrand over them to find our result. Now, the result he attained was found by splitting up the absolute value inequality (setting the integrand greater than zero since we are supposed to maximize) into 4 cases and by symmetry solving a DOUBLE integral multiplied by 4 to compute the answer. I don't understand how it became a double integral. My gut told me that the answer was infinity(this is my attempt at a solution, I don't see how it doesn't equal this result) because there aren't any implied bounds for the z axis in the given function. The only problems like this that I have seen before always involved an integrand including all three variables that usually resulted in a sphere as your bounds of integration. I will be asking him about it today but I thought I'd see what others had to say too.

 

[LCKurtz]

My guess is that the problem is misprinted and intended to be a double integral in R². You would integrate over the square |x|+|y| ≤ 1 to maximize the integral, just as you seem to think.

 

[Lavoisier44]

Thanks for the reply. That was my guess too. He has made mistakes in his problems before so...