How to Calculate the Mass of a Pyramid Using Triple Integrals?

[1MileCrash]

Homework Statement

Find the mass m of the pyramid with base in the plane z = 9 and sides formed by the three planes y = 0 and y - x = 5 and 6x + y + z = 28, if the density of the solid is given by δ(x,y,z) = y.

Homework Equations

The Attempt at a Solution

This problem is driving me insane. It takes me about 45 minutes of algebra to evaluate this incorrectly set up integral..

I integrated y in the order dz dy dx, limits, respectively:

9 to 28-6x-y
0 to 5+x
0 to 2

Is that correct? I don't really know how to get the limits for x.. this is so hard to picture in my mind!

To get 2, I solved the system 6x + y + z = 28 with z = 9 and y = 5 + x, and for 0, I just guessed.

Help please!

 

[OldEngr63]

May I suggest that you draw a sketch and get a clear grip on the limits before you try to evaluate anything.

 

[1MileCrash]

Well, that's what I'm trying to do..

 

[1MileCrash]

I drew it again, the only thing different is that my order is dzdxdy.

I don't understand how to picture these things, it's very difficult.. I just drew an xy plane noting that z = 9 for the base, and looked at that. What would you do?

 

[1MileCrash]

I think this is far beyond me.

 

[1MileCrash]

Nevermind, found a good explanation online of something called the "shadow method."

 

[LCKurtz]

Have you figured out it gets tricky if you use any order of integration other than dzdxdy?