Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

290 0 CHAPTER 15 MULTIPLE INTEGRALS
6. A(S) = JJ 0 J [f.,(x, y))2 + [f11(x, y))2 + 1 dA
(b) We usually evaluate f[JB f(x, y , z) dV as an iterated integral according to Fubini's Theorem for Triple Integrals
(see Theorem 15.7.4).
(c) See the paragraph following Example 15.7.1.
(d) See (5) and (6) and the accompanying discussion in Section 15.7.
(e) See (10) and the accompanying discussion in Section 15.7.
(f) See (11) and the preceding discussion in Section 15.7.
8. (a) m = JJJE p(x, y, z) dV
. (b) M 71z = JJJE xp(x, y, z) dV, Mxz = JJJE yp(x, y, z) dV, Mxy = JJJE zp(x, y, z) dV.
· (- - - ) J - Myz _ M xz d _
( c ) Th e cen
lvfx
t er o f mass ts x, y, z w 1ere x = --, y = --,an z = --. 11
m m m
9. (a) See Formula 15.8.4 and the accompanying discussion.
(b) See Formula 15.9.3 and the accompanying discussion.
(c) We may want to change from rectangular to cylindrical or spherical coordinates in a triple integral if the region E of
integration is more easily described in cylindrical or spherical coordinates or if the triple integral is easier to evaluate using
cylindrical or spherical coordinates.
a (x, y) I ax;au ax;av I ax ay ax By
10· (a) a (u, v) = ay;au ayjav = au av - av au
(b) See (9) and the accompanying discussion in Section 15.10.
(c) See (13) and the accompanying discussion in Section 15.10.
TRUE-FALSE QUIZ
1': This is true by Fubini's Theorem.
3. True by Equation 15.2.5.
5. True. ~y Equation 15.2.5 we can write J~ f0 1 f(x) f(y) dy dx = I~ f (x) dx I~ f(y) dy. But J~ f(y) dy = J~ f(x) dx so
this becomes I; f(x) dx .{ 0
1
f(x) dx = [J; f(x) dxf.
7. True: J 0
J 4 - x 2 - y 2 dA = the volume under the surface x 2 + y 2 + z 2 = 4 and above the xy-plane
= ~ (the volume ofthe sphere x 2 + y 2 + z 2 = 4) = ~ · t1r(2) 3 = 1 36
1r
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