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**Unformatted text preview: **Math 21C
Kouba Discussion Sheet 6 1.) Let R be the solid region bounded by the surfaces .2 2 V4 —- m2 — y2 and z = 0 . SET
UP BUT DO NOT EVALUATE triple integrals which represent the volume of the solid
a.) using rectangular coordinates.
b.) using cylindrical coordinates.
c.) using spherical coordinates. 2.) Let R be the solid region bounded by the surfaces 2 = «$2 + 3/2 and z = V 18 — m2 -— 3/2
. SET UP BUT DO NOT EVALUATE triple integrals which represent the volume of the solid
a.) using rectangular coordinates.
b.) using cylindrical coordinates.
c.) using spherical coordinates. 3.) Let R be the solid region inside the surface 5102 + y2 = 4 and bounded by the surfaces 2 = 0 and z : V9 — 3:2 — 3/2 . SET UP BUT DO NOT EVALUATE triple integrals which
represent the volume of the solid a.) using rectangular coordinates.
b.) using cylindrical coordinates.
c.) using spherical coordinates. 4.) Consider the UFO bounded by the surfaces z = $2 + y2 and z = 8 — m2 — y2. The
density of the UFO at point P = (x, y, z) is given by the square of the distance from P to
the z—axis. SET UP BUT DO NOT EVALUATE triple integrals in all three coordinate
systems which represent the UFO’s a.) average density. b.) total mass.
c.) total volume. d.) z—coordinate of the centroid.
e.) z—coordinate of the center of mass.
f.) moment of inertia about i.) the z-axis. ii.) the X—axis. 5.) Convert the following integral to rectangular coordinates. DO NOT EVALUATE THE
INTEGRAL. 1r/2 77/4 cos¢
/ / / p3 sin 0 sin2 45 dp qu d0
0 O 0 6.) SET UP BUT DO NOT EVALUATE a triple integral which represents the volume of
the given doughnut (torus). 7.) Consider the solid region R bounded by the surfaces 2 = m2 + y2 and z : 2y. Describe
B using spherical coordinates. THE FOLLOWING PROBLEM IS FOR RECREATIONAL PURPOSES ONLY. 8.) The camp cook wants to measure exactly four ounces of vinegar out of a jug, but has only a ﬁve—ounce container and a three—ounce container. How can the cook accomplish the
task ? ...

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