tripleintegrals

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

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

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