1. ## iterated integrals

Hey guys can you help me with the following problem plz...

Consider the tetrahedron bounded by the plane 4x + 8y + z = 16 and the 3 coordinate planes.

Give the iterated integrals for the volume of the tetrahedron which uses the dz dx dy order of integration, then the dy dz dx order.

Any help is greatly appreciated!

2. Originally Posted by boousaf
Hey guys can you help me with the following problem plz...

Consider the tetrahedron bounded by the plane 4x + 8y + z = 16 and the 3 coordinate planes.

Give the iterated integrals for the volume of the tetrahedron which uses the dz dx dy order of integration, then the dy dz dx order.

Any help is greatly appreciated!
It's essential to draw a diagram. Get the edges that lie in each plane as follows:

In the xy-plane, z = 0 and so the equation of the edge is x + 2y = 4.

In the zy-plane, x = 0 and so the equation of the edge is z + 8y = 16.

In the zx-plane, y = 0 and so the equation of the edge is 4x + z = 16.

Then it's not hard to see what the integral terminals will be for a particular order of integration.

Eg: The order dz dx dy.

The integral terminals are:

z = 0 (lower plane) to z = 16 - 8y - 4x (upper plane).

And the region of integration in the xy-plane is given by

x = 0 to x = 4 - 2 y

y = 0 to y = 2

Eg. The order dy dz dx.

y = 0 (lower plane) to y = (16 - 4x - z)/8 (upper plane)

And the region of integration in the zx-plane is given by ......

3. thanks mr fantastic, I see it now