MATH 23 – Midterm 2 Fall 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 60.
1. (10 pts: 5 each) The mass Mof a thin plate Din the xy–plane can be calculated as
M=Z1
0Z2√x
0
ρ(x, y)dy dx,
where ρ(x, y)is the density function.
(a) Sketch the shape of this thin plate D.
(b) Change the order of integration.
2. (15 pts: 5 each) A solid Vlies above the cone z=px2+y2and below the sphere x2+
y2+z2= 2. Set up, but do not evaluate, iterated integrals to find the volume of Vin the
following coordinate systems.
(a) The Cartesian (rectangular) coordinates.
(b) The cylindrical coordinates.
(c) The spherical coordinates.
3. (10 pts: 5 each) ~
F(x, y) = ycos x
~
i+ sin x~
jis a conservative vector field.
(a) Find a potential function ffor ~
F. (That is, find a scalar function f(x, y)such that ~
F=
∇f.
(b) Use the Fundamental Theorem for Line Integrals to calculate RC~
F·d~r, where Cthe the
part of the curve y= sin xgoing from (0,0) to (3π/2,−1).
CONTINUED ON THE BACK −→
1
