Lecture32and33

# The top and bottom meanwhile may curve in two

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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 (2-D) 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 re...
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## This document was uploaded on 03/30/2014.

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