MATH 23 – Midterm 2 Spring Semester 2007
Duration: 50 minutes
Instructions: Answer all questions, without the use of notes, books or calculators. Partial credit will be
awarded for correct work, unless otherwise specified. The total number of points is 100.
1. (20 pts) Consider the domain Rin the xy-plane such that 0≤y≤1and 0≤x≤2and x2+y2≥4.
(a) Draw this domain.
(b) Setup 2 integrals to evaluate the volume between a function f(x, y)>0and the plane z= 0 over
R, one integrating xfirst and the other integrating yfirst.
(c) Evaluate the volume above Rand below the surface z=xy.
2. (18 pts) A drill bit is shaped like the inside of the cone z2=x2+y2for 0cm≤z≤2cm. Write down an
integral describing the mass of this drill bit if its density is d(x, y, z) = 300
x2+y2+2z2+100 g/cm3in TWO
of the following THREE coordinate systems.
(a) Cartesian coordinates.
(b) Cylindrical coordinates.
(c) Spherical coordinates.
3. (20 pts) Consider the force field ~
F(x, y) = 1/x ~
i+ 2/y ~
jand the curve C, a parabola going from (1,1)
to (3,9) via the point (2,4)
(a) Compute the work of ~
F(x, y)done on a particle traveling along Cby parametrizing the curve.
(b) Find the potential φ(x, y)such that ∇φ=~
Fand use it to verify your answer to part a).
(c) Could you use Green’s theorem to find the work done by ~
Fon a particle going counterclockwise
around the circle of radius 1 centered at the origin? Justify your answer.
4. (18 pts) Consider the lower half of the sphere of radius 2m centered at the origin
(a) Parametrize the surface described above.
(b) Compute the flux of ~
F= (1/x~
i+ 3/y ~
j+x2~
k)m/min through the half sphere oriented outward.
(c) Assume the flux of ~
Fthrough a surface Sis 2m3/s. If the surface describes a gold-digger ’s pan
and ~
Fis the velocity of water containing 0.001 ounces of gold per meter cubed, how long would
it take to gather 4 ounces of gold?
5. (24 pts) Answer the following questions in no more than two lines of text or formulas (much less is
usually needed if you are right on point).
(a) What is a formula for the average height of a surface z=f(x, y)over a domain Rin the xy-plane?
(b) If d(x, y)is the density of fairy shrimp per unit length of a waterway, what does RCd(x, y)dl
represent if Cis the path of the waterway?
(c) For a given velocity field ~
F(x, y, z), how would you orient a surface Sto maximize the flux
through that same surface?
(d) Sketch or describe a 2-dimensional vector field, ~
F, for which curl ~
F= 0 inside the circle of radius
1 centered at the origin and curl ~
F=−1everywhere else.
(e) Sketch or describe the curve parametrized by
x=t, y =e−tcos t, z =e−tsin t
for 0≤t≤ ∞.
(f) If ~r(s, t) = x(s, t)~
i+y(s, t)~
j+z(s, t)~
k, is the parametrization of a certain surface, what can you
say about the length and direction of the vector (~rt×~rs)∆s∆t?
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