18.022 Recitation Quiz (with solutions)
01 December 2014
1. Let S be the surface defined by
z=
1
for z ≥ 1.
x2 + y2
(a) Sketch the graph of this surface.
(b) Determine whether the volume of the region bounded by S and the plane z = 1 is finite or
infinite.
(c) Determine whether the surface area of S is finite or infinite.
Solution. (a) See the graph below (cut off at z = 10).
R 1R 2
R 1R 2
R1
(b) The volume is given by 0 0 π(r−2 − 1)r dθ dr = 0 0 π(r−1 − r) dθ dr. This is infinite since 0
R1
is infinite and 0 r dr is finite. (c) The surface area of S is given by
Z
0
1 Z 2π
0
Z
q
2
2
1 + fx + f y r dθ dr =
1 Z 2π
0
Z
=
0
1 Z 2π
0
0
1
Z
= 2π
dr
r
q
1 + x2 /(x2 + y2 )4 + y2 /(x2 + y2 )4 r dθ dr
√
1 + r−6 r dθ dr
√
r2 + r−4 dr.
0
This integral is infinite because the second term dominates, and
rigorously, we can drop the first term:
Z
2π
0
1
√
1
Z
r2 + r−4 dr ≥ 2π
0
R1
dr
0 r2
= +∞. To prove this
√
r−4 dr = +∞.