MATH 23 – Practice 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 1≤x≤2 + cos yand 0≤y≤π.
(a) Draw this domain.
(b) Set up 2 integrals to evaluate the volume between a function f(x, y)>0and the plane z= 0
over R: one integrating xfirst and the second integrating yfirst.
(c) Evaluate the volume above Rand below the surface z=exsin y.
2. (18 pts) A spherical orange of radius 4cm and with center at the origin, is sliced in eight equal parts by
cutting it vertically along the xand yaxis and along the planes y=xand y=−x. Write the integral
you would use to compute the mass of the slice of orange between the y=xplane and the y-axis if
the density of the orange is d(x, y, z) = 0.5+0.2(x2+y2)/(z2+ 1).
(a) In cylindrical coordinates.
(b) In spherical coordinates.
3. (20 pts) Consider the vector field ~
F=x2y3
~
i+ 0~
jand the curve Cdescribing the boundary of a square
of side 2 centered at the origin with sides parallel to the axes.
(a) Compute the line integral of ~
Fover Cby parametrizing the curve.
(b) Can you use Green’s theorem to compute this line integral? Why or why not?
(c) If you can use Green’s theorem, do so, if not suggest a simple modification to the problem that
would allow you to use it.
4. (18 pts) Consider the cylinder of radius 3 centered on the x-axis for 0≤x≤2.
(a) Draw the cylinder and parametrize its surface.
(b) Compute the flux of ~
F=yz~
i+xy~
j+xz~
kinto that cylinder.
(c) If ~
Fdescribe the velocity field of flowing water in m/s and the cylinder has radius 3m, what
does the previous calculation describe, in non-mathematical terms?
5. (24 pts) Answer the following questions in no more than two lines of text (much less is usually needed
if you are right on point).
(a) If a vector field ~
Fis such that its circulation around any closed loop is 0, what can you say about
the line integral between two points P= (x0, y0)and Q= (x1, y1)along a straight line compared
to that between the same points along a path twice a long?
(b) When do you use the Jacobian ∂(x,y)
∂(s,t)of a transformation x(s, t)and y(s, t)from the (x, y)coordi-
nates to the (s, t)coordinates?
(c) Describe or sketch the surface parametrized by x=s,y=ssin t,z=scos t.
(d) If ~
Fis the velocity field of the wind and the air contains 0.1 grams of pollen per meter cubed,
how much pollen would you find in a surface Safter 30 minutes if R RS~
F·~ndA = 6m3/min?
(e) Find a 3-dimensional vector field ~
F(x, y, z)where each vector has length 2 and points towards
the point (2,3,4) (except at (2,3,4) where ~
F(2,3,4) = 0
~
i+ 0~
j+ 0~
k).
(f) Write a formula to compute the average value of g(x,y , z)over a three dimensional domain V.
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