Question

Can someone help me decide on the correct bounds of integration for this triple integral?

Find the volume of the pyramid with base in the plane z=-5, and sides formed by the three planes y=0, y-x=4, and x+2y+z=4

Answer 1

I'm just having trouble visualizing what the region is... I've tried
\[\int\limits_{-9}^{-4} \int\limits_{0}^{x+4} \int\limits_{-5}^{4-x-2y} (1) dz dy dx\]
to no avail. I feel like the y and z bounds are good but I just couldnt figure out the x bounds.

Once I get the bounds I should have no problem solving... Thanks in advance!

Answer 2

Even with technology, it's still quite difficult to visualize -.-

Answer 3

I know... o.o' I've tried wolfram, a random 3D graphing java application and old fashioned pen and paper... I suppose the vertices would occur when 3 of the equations coincide so i've messed with finding those a bit... still havent made too much progress...

Answer 4

first draw the cross section of base in z=-5 plane