Find the volume V of the solid in the �rst octant bounded by y = 0, z = 0, y = 3,

Question

z = x, and z + x = 4.

The best part about this problem is that I solved it using basic geometry but am expected to do it using a triple integral that will take up a whole page.

I'm honestly lost on how im supposed to set up the integral I've tried setting it up various ways and i never really even come close to the volume 12, which it is.

Answer 1

Looks like a triangular prism, with base a 3x4 rectangle in the xy plane, and top edge a line parallel to the y-axis at x=2, z=2.

I also get v=12 using simple geometry.

Integrating in two pieces, one under z=x, and the other under z=4-x, we get

∫[0,3]∫[0,2]x dxdy + ∫[0,3]∫[2,4](4-x)dx dy
= ∫[0,3](1/2 x^2)[0,2]dy + ∫[0,3](4x-x^2/2)[2,4]dy
= ∫[0,3]2dy + ∫[0,3]2dy
= 6+6
= 12