The top and bottom meanwhile may curve in two

Info iconThis preview shows page 1. Sign up to view the full content.

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

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...
View Full Document

This document was uploaded on 03/30/2014.

Ask a homework question - tutors are online