Extra Credit Problems

Math 166

Fall 2000

1. Consider the function where means the greatest integer less than or equal to . Find a closed form formula for (without an integral sign). Is continuous? Find Is ? Does this contradict the fundamental theorem of calculus?
2. Let be differentiable and let be continuous. Show that if we define , then

3. Show that the centroid of any triangle occurs at the intersection of the median lines.

4. a) Consider the function Show that the area bounded by this curve and the x-axis is finite. Compute the centroid of the region bounded by this curve and the x-axis. Find the volume obtained when the region bounded by this curve and the x-axis is revolved about the x-axis. Find the volume when this region is revolved about the y-axis.

b) Is it true that a region with finite area must have a centroid? (This is true for a compact region…that is a region (which we will say is bounded by continuous functions) that can be put inside a circle of big enough radius…show this fact). If the answer to the first question is “no” see if you can give conditions under which a region of finite area possesses a centroid. (Actually I cannot resist pointing out that the x coordinate of the centroid of the region from a) is "infinity", hence even though the region has finite area, you cannot "balance" it with any vertical stick. You can, however, balance it with a horizontal stick...generalize).

5. I have a can of radius R and height h that is made of metal of uniform density r. In the can is some liquid swill of uniform density s. When the can is full, then center of mass is “dead center”. This is also true when the can is empty. Find out when the can is the most stable (that is, when the center of mass is lowest).

6. We know that if you can rearrange a conditionally convergent series so as to make it add up to any real number that you want (you can also rearrange it so that it diverges). Under what conditions can you rearrange a divergent series and obtain a convergent one?