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Unformatted text preview: right sides are curved in one dimension only, meaning that
you could easily mold a sheet of paper to them with no crumpling. The top and bottom,
meanwhile, may curve in two dimensions (they are unrestricted surfaces ). The integral of a Figure 2:
function f over such a domain can be expressed as
ˆ D f (x, y, z ) dV = ˆ bˆ
a φ2 (x) ˆ γ2 (x,y ) φ1 (x) f (x, y, z ) dzdydx. (1) γ1 (x,y ) To understand why it must have this form, consider that a deﬁnite integral is a number, so
the last pair of limits of integration we use must be constants. That is, the outer integral must
be the integral in x, from x = a to x = b. Now, for each value of x in the interval [a, b], the
values of y run from y = φ1 (x) to y = φ2 (x). These limits are constant with respect to the
remaining variables y and z , so we can make this our middle integral. Finally, for each point
(x, y ) in the region labelled R, the values of z in the domain D run from z = γ1 (x, y ) up to
z = γ2 (x, y ).
It might help to view a triple integral as a double integral of a single integral1 . Viewed this
ˆ γ2 (x,y)
way, Equation (1) is a double integral over R of the function g (x, y ) =
f (x, y, z ) dz .
γ1 (x,y ) Setting up a double integral should be familiar now; R is a Type I (2D) region, so we write
ˆ b ˆ φ 2 ( x)
g (x, y ) dydx. Of course, for this to be helpful we need to understand what the
a φ 1 ( x) function g represents: it can be thought of as the sum of all the values of f along a vertical
1 You can also view it as a single integral of a double integral, if you prefer! 2 column, one of which extends above each point (x, y ) in the region R from z = γ1 (x, y ) up to
z = γ2 (x, y ). See Figure 3. Figure 3: Example 2: Evaluate ´ D xz dV , where D is the region in the 1st octant (where x, y , and z are all positive) below the plane z = y and inside the cylinder (see Figure 4; the second
image is a view from a diﬀerent perspective, with the extra sections of the plane and cylinder
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This document was uploaded on 03/30/2014.
 Winter '13
 Calculus, Integrals

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