MA 230
Problem of the Day
January 15, 2003
Let 1 be the line through the points (1, 0, 0) and (1, 2, 2), and let 2 be the line through
the points (1, 1, 1) and (1 + 2, 2, 2). Do 1 and 2 intersect? If so, at what point do they
intersect?
MA 230
Problem of the Day
February 3, 2003
Let
f (x, y ) = y cos x.
Find the points on the graph of f where the tangent plane is parallel to the plane
x 3 y + 2z = 2.
MA 230
Problem of the Day
Consider the solid region S under the plane
z=
y
3
and above the rectangle
R = cfw_(x, y ) | 0 x 2 and 0 y 3.
Use Cavalieris principle to compute the volume of S in two dierent ways:
1. using slices by planes where x is constant,
MA 230
Problem of the Day
Calculate the average value of the function
f (x, y ) = xexy
over the rectangle
R = cfw_(x, y ) | 0 x 1 and 0 y 2.
March 19, 2003
MA 230
Problem of the Day
March 24, 2003
Let Q be the region in R3 bounded by the cylinder x2 + z 2 = 9, the plane y + z = 3, and the
plane y = 0. Calculate
z dV.
Q
MA 230
Problem of the Day
March 26, 2003
Determine the formula T (x, y ) for the mapping T : R2 R2 that corresponds to rotation of
R2 about the point (2, 1) by 90 in the counterclockwise direction.
MA 230
Problem of the Day
March 28, 2003
Use spherical coordinates to compute the volume of the solid that is inside the sphere
x2 + y 2 + z 2 = 5
and outside the cone
z 2 = 3x2 + 3y 2 .
Another way of saying outside the cone is
z 2 3x2 + 3y 2 .
MA 230
Problem of the Day
March 31, 2003
Compute the center of mass of the region in the rst octant enclosed by the sphere
x2 + y 2 + z 2 = 4
assuming that the density is constant.