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Unformatted text preview: Triplelntegrals http://www.mautexasedu/users/gilbert/asciimathfﬁiplelntegralshtml ENEEGRACHGN in several variables: TRIPLE ENTEGMLS Once we've seen the pattern of how to go from integrals of functions of one vmiable to double integrals of ﬁmctions of two
variables, then studying triple integrals if EfOc, ya 3} dxdydg of functions f(x, y, z) of three variables is a natural step, and a very important one in appiications. Now 9 the domain of integration is a solid E in 3—space, and Q when f > 0 the integral is interpreted as the voiume of the solid in 4«space below the graph of f and above E; just don't expect lots of graphics of things in 4wspacel As v'vith double integrals, having a detailed understanding of the domain of integration is the key to setting up a triple
integral. But for the moment let‘s simply see how those essential ingredients will enable us to evaluate a triple integral: ﬁrst
ﬁx a point (x, y) in the xywplane, and introduce the ﬁniction 990'. y) VD; y) = f ftx, y, as; W. y) then, if @(x, y), 1/106, y) are chosen appropriately, the tripie integral reduces to a repeated integral ffﬁqﬂx, y, z)a’xdydz "w" IL) V(x, y)dxdy = Hrfifﬁ: ﬁx, y, z) dz)dxdy for the right choice of domain D in the xy—piane. In fact, if we remember that the main step in evaluating a double integral
was reducing it to repeated integrals of one variabie, we see that If L f(x, y, z)dxdydz 1 £1 f::)( f::::: ﬁx: )2, z) dzjdy) dx, which illustrates well the iciea all along: calculus of ﬁmctions of several variables proceeds by reducing it to the caicuius of functions of one variable. For triple integrals this amounts to making the right choices of limits of integration! Examples
Show how: 10f3 11/30/2009 4:53 PM Triplelntegtals http://www.ma.u1exas.edufusers/ giibert/asciimam/Triplelntegtelslitml memMWWWmemmmm . ﬁlm Question hexpress the triple integral I m H E m, y, z)dxdydz t
I
s
L
l
l .2 l L as a repeated integral when E is the solid above the sayplane
and below the graph of z e 6 ~ y. g So E is the familiar solid whose cross—sections are
E congruent trapezoids. But here we look at the solid differently:
‘ﬁx (x, y) in the base, tie. a S x S I), while e 5y 5 d, and let z I
i go vertically ﬁom the point (x, y) up to the top of the solid, _' ten, 0 E z S 6  y. Thus with the previous notation, . l l ; D * {maximal}, WUQJ’) r" 6w,
i in which case I e IXET :_y ﬁx, y, z)a’z)dy]dx. Here's a slightly more complicated solid. mwmwfmmmlmmmmmmwmwmw  g G Question 2: express the triple integral 1 = [HF ﬁx, 3:, z)a’xa’ya’z as a repeated integral when E is the solid above the xy
~plane bounded by the graphs of
y m 4):2, z = y. Notice ﬁrst that the surface graph ofy : 4 W x2 is a
parabolic cylinder, while the graph of z = y is a plane 1
passing through the x—axis and intersecting the yz~plane in the
: line 2 z y. So if E lies above the xy—plane and is bounded by
,5 this parabolic cylinder as well as the plane 2 r» y it will look
j like the solid to the right. ' We have to determine the limits of integation. Now D is a
i region in the xy—plane that can be found from the overhead ' View of E. Since the parabola y = 4 w x2 intersects the x—axis
at x m d: 2, the region D is shown below to the right. Thus E
‘ consists of all points (x, y, 2) such that V. mmmmmmnn V 052$y,05y£4it“, w2sx52. so as a repeated integral, 1 = f 1]:ng ﬂx, y, z)dz)dy]a’x. Question 3: the order of integration for integrating over D could have been reversed so that 2 Of3 11/30/2009 4:53 PM Triplefntegrals http:f/www.ma.atexas.edu/users/giIbert/asciimafh/Tréplekltegrals.html I " WWW” a a )4 L;
L‘ (10:) W(x.)’)  x: y, :a} ,5 A 'y Use the previous graph of D to determine City), My), c, and d. In fact, it may be necessary to integrate ﬁrst with respect to x or )2. That will depend on what the domain of integratiorx
E looks like. The next question illustrates this WWW WWIWWWWWWW Question 4: express the integral I a ff E ﬁx, y, z)dxdydz I g as a repeated integral I = fiUfUiiliZ m Madge E
E
E
E
E when E is the solid Shawn to the J fghr lying to the right ofE
E the xz—plane and bounded by the graphs of
E 3 of3 11/30/2009 4:53 PM ...
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 Fall '09
 Gilbert

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