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Lecture32and33 - 10 Triple Integrals If youve understood...

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10 Triple Integrals If you’ve understood the preceding discussion, then the concept of a triple integral should be a natural extension of the concept of (single) integrals and double integrals. We will now be speaking of functions of three variables, and our domains of integration will be three- dimensional. As we’ve done for double integrals, we’ll avoid rigorous deFnitions, and instead give outlines which will (we hope) enable you to construct the integrals you need. If the domain of integration (call it D ) is a rectangular box, with x [ a 1 ,a 2 ] ,y [ b 1 ,b 2 ] ,z [ c 1 ,c 2 ] , then the jump from two variables to three is an easy one. We divide all three axes into intervals of lengths x , y ,and z . In each of the resulting small boxes we choose a point ( x i j k ) , evaluate the function (let’s call it f )atthatpo in t ,andmu lt ip ly the result by the volume of the box V = x y z (see ±igure 1). We then sum the results ±igure 1: for every one of these boxes within the domain. The limit of the result as x, y, z all approach zero is our integral, ˆ D f ( x, y, z ) dV . Of course, there are now six possible orders of integration, and for rectangular domains these can all be interchanged freely: ˆ D f ( x, y, z ) dV = ˆ c 2 c 1 ˆ b 2 b 1 ˆ a 2 a 1 f ( x, y, z ) dxdydz = ˆ c 2 c 1 ˆ a 2 a 1 ˆ b 2 b 1 f ( x, y, z ) dydxdz = ˆ b 2 b 1 ˆ c 2 c 1 ˆ a 2 a 1 f ( x, y, z ) dxdzdy &c. . ±or domains which are not rectangular, we can extend the concept of “Type I”, “Type II”, and “Type III” regions. .. but we can now distinguish between at least seven types! Numbering them isn’t necessary; if we grasp the concept we should be able to write down an appropriate integral.
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Example 1: Suppose the domain of integration can be described as D = { ( x, y, z ) | a x b, φ 1 ( x ) y φ 2 ( x ) , γ 1 ( x, y ) z γ 2 ( x, y ) } , as shown in Figure 2 (this is the three-dimensional “Type I” region). This isn’t easy to show with a static 2-D image, but what we’re describing is a solid whose front and back sides are ±at (they are in the planes x = a and x = b ), and whose left and 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 b a ˆ φ 2 ( x ) φ 1 ( x ) ˆ γ 2 ( x,y ) γ 1 ( x,y ) f ( x, y, z ) dzdydx. (1) To understand why it must have this form, consider that a deFnite 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
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