Math 113 – Calculus III Chapter 16 Practice Problems Fall 2003
1. Suppose the integral of some function fover a region Rin the plane is given in polar coor-
dinates as
Z3
0Zπ
2
0
r2dθdr.
(a) Sketch the region of integration Rin the xy plane.
(b) Convert this integral to Cartesian coordinates.
(c) Evaluate the integral. (You may use either polar or Cartesian coordinates.)
2. Let Wbe the solid region where x≥0, y≥0, z≥0, z≤x+y, and x2+y2≤4. (In other
words, Wis bounded by the yz plane, the xz plane, the xy plane, and the surfaces z=x+y
and x2+y2= 4.)
Let f(x, y, z) = 1 + x+ 2zbe the density of the material in this region.
Express the total mass of the material in Was a triple integral in
(a) rectangular coordinates,
(b) cylindrical coordinates.
Your expressions should be complete enough that, in principle, they could be evaluated, but
not evaluate the integrals!
3. The following triple iterated integral uses spherical coordinates.
Zπ/2
0Zπ/4
0Z2/cos φ
1/cos φ
ρ2sin φ dρdφdθ
Sketch (and describe) the region of integration.
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