Multivariate Calculus - Bruce E. Shapiro

Lecture 20
Triple Integrals
Triple Integrals in Cartesian Coordinates
We can easily extend our definition of a double integral to a triple integral. Suppose
that f(x, y, z) : V ∪ R 3 ↦→ R. Then we extend our Riemann sum so that it covers
the volume V with small boxes V j of dimension
The volume ∆V j of the ith box is then
∆x j × ∆y j × ∆z j
∆V j = ∆x j ∆y j ∆z j
Then the Riemann Sum representing ∭ V
f(x, y, z)dV is
n∑
V ≈ f(x j , y j , z j )∆V i
j=1
and the triple integral of f over V is
V
f(x, y, z)dV =
lim
n→∞,∆V j →0
j=1
n∑
f(x j , y j , z j )∆V i
We can also make general definitions of x-simple and y-simple sets. These definitions
are exactly the same as they were previously but with an added dimension in the
domain.
Example 20.1 Find
∫ 2 ∫ 3 ∫ 2x−y
0 −2 0
dzdydx
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