tripleintegrals

# tripleintegrals - Math 21a Handout on Triple Integrals The...

This preview shows pages 1–2. Sign up to view the full content.

Math 21a Handout on Triple Integrals The purpose of this handout is to provide a few more examples of triple integrals. In particular, I provide one example in the usual x-y-z coordinates, one in cylindrical coordinates and one in spherical coordinates. Example 1 : Here is the problem: Integrate the function f(x, y, z) = z over the tetrahedral pyramid in space where 0 x. 0 y. 0 z. x + y + z 1. (1) The integral in question is I zdz dA xy R0 1 −− ∫∫ , (2) where R is the ‘shadow’ region in the x-y plane; the region where 0 x, 0 y, x + y 1. (3) Indeed, a vertical line (where x and y are constant) will hit the pyramid only if x and y are non- negative and x + y 1. Otherwise, one of the conditions in (1) is violated when z 0. This line enters our pyramid from below where z = 0 and it then exits with z value where x + y + z = 1, which is to say where z = 1 - x - y. This information provides the lower bound to the z-integral, 0, and also the upper bound, 1 - x - y.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## tripleintegrals - Math 21a Handout on Triple Integrals The...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online