5.
Triple
Integrals
5A.
Triple integrals in rectangular and cylindrical coordinates
~d~
lx
5A1
Evaluate: a)
b)
l2
'
2xy2z dz dx dy
5A2.
Follow the three steps in the notes to supply limits for the triple integrals over the
following regions of 3space.
a) The rectangular prism having as its two bases the triangle in the yzplane cut out
by the two axes and the line y
+
z
=
1,
and the corresponding triangle in the plane x
=
1
obtained by adding
1
to the xcoordinate of each point in the first triangle. Supply limits
for three different orders of integration:
(iii)
///
dy dx dz
b)* The tetrahedron having its four vertices at the origin, and the points on the three
axes where respectively x
=
1, y
=
2, and z
=
2. Use the order
///
dz dy dx.
c)
The quarter of a solid circular cylinder of radius
1
and height 2 lying in the first
octant, with its central axis the interval
0
5
y
5
2 on the yaxis, and base the quarter circle
in the xzplane with center at the origin, radius
1,
and lying in the first quadrant. Integrate
with respect to y first; use suitable cylindrical coordinates.
d) The region bounded below by the cone z2
=
x2
+
y2, and above by the sphere of
radius
4
and center at the origin. Use cylindrical coordinates.
5A3
Find the center of mass of the tetrahedron
D
in the first octant formed by the
coordinate planes and the plane x
+
y
+
z
=
1. Assume 6
=
1.
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 Spring '08
 Auroux
 Integrals, Limits, Mass, Fundamental physics concepts

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