The following write-ups are due Monday May 1.

Monday April 24
Solve, write up and turn in:

Sec. 20.4: 39 a, g, 40
Calculate the integral on the whole real line of the exponential of  -x^2+2x
(Hint: complete the square.)

Tuesday April 25
Solve, write up and turn in:

Two good practice problems. You can check your answers using formulas from geometry.

1. Set up a triple integral to calculate the volume of the cylinder  bounded by x^2+y^2=4, z=0, z=3.
Calculate this integral in two ways:
   a) Set up horizontal slicing (parallel to the xy-plane).
   b) Set up slicing parallel to the xz- plane.



2. Set up a triple integral to calculate the volume of the cone bounded by
      cone
Calculate this integral in two ways:
   a) Set up horizontal slicing (parallel to the xy-plane).
   b) Set up slicing parallel to the xz- plane.

Sec. 20.5: 11, 15, 17, 19 (This is like the problem we were solving in class. To see the limits for x,
it may be helpful to sketch the region Ry separately, in an xz- plane.)

Wednesday April 26

Solve the problems listed on yesterday's assignment.

Thursday April 27
Solve, write up and turn in:
Sec. 20.6: 1, 2, 11, 12, 15

Friday April 28

Note on problem 20.6 #12: the limits for theta: indeed, it is clear from the picture that theta is between -pi/2 and pi/2.
Also, for an argument using algebra: the inequality containg theta is  r^2<z<2r cos(theta) so  cos(theta)>r/2>0 so cos(theta)>0.

Note on problem 20.6 #11: the text of the problem is not clear enough. It appears that the question is
to calculate the total volume removed from the sphere (rather than "of the cylindrical hole", which is too trivial).

Bonus problem (extra 6 points). Consider a domain D in the xy-plane, and a function f(x,y) defined on D.
Consider the region R bounded by the graph z=f(x,y) of f and the xy-plane. If f(x,y)>0 on D then we can calculate the volume of R in two ways:
as the double integral of    f(x,y) dx dy  on D, or as a triple integral of   dx dy dz   on V.
a) Show that the two numbers are equal.
b) What happens if f(x,y)<0 on D?
Today we learned about moments of inertia about an axis (Sec. 20.6),
about center of mass, centroid (see formulas (7,8) in 20.3 for 2-d and p. 732 in 3-d).
Solve, write up and turn in:
Sec. 20.3 :  3,  4, 5
Sec. 20.6:  2a, 4, 6