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

276 0 CHAPTER 15 MULTIPLE INTEGRALS
1 L2 L2 L2 La
£ 3·222 = 8
55. (a) The triple integral will attain its maximum when the integrand 1 - x 2 - 2y 2 - 3z 2 is positive in the region E and negative
everywhere else. For if E contains some region F where the integrand is negative, the integral could be increased by
excluding F from E, and if E fails to contain some part G of the region where the integrand is positive, the integral could
be increased by including Gin E. So we require that x 2 + 2y 2 + 3z 2 $ 1. This describes the region bounded by the
ellipsoid x 2 + 2y 2 + 3z 2 = 1.
(b) The maximwn value of JjJE (1 - x 2 - 2y 2 -
3z 2 ) dV occurs when E is the solid region bounded by the ellipsoid
x 2 + 2y 2 + 3z 2 = 1. The projection of E on the xy-plane is the plaf1ar region bounded by the ellipse x 2 + 2y 2 = i, so
and
using a CAS.
15.8 Triple Integrals in Cylindrical Coordinates
.1. (a) From Equations I, x = r cos 9 = 4 cos i = 4 · ~ = 2,
y '= rsinfJ = 4sin ~ = 4 · ~ = 2.J3, z = ~ 2,
3
so the point is
X
I
- 2 •
I
l (4, 3. -2)
Y · (2, 2.J3, -2) in rectangular coordinates ..
(b)
(2.-¥. 1)
x = 2cos(-~) = 0, y = 2sin( - ~l) = - 2,
and z == 1, so the point is (0, - 2, 1) in rectangular coordinates.
X
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