Solutions should be fully derived showing all intermediate results, using class procedures. Show all reasoning. Bare answers are absolutely not acceptable, because I will assume they come from your calculator (or the math handbook, sometimes,) instead of from you. You must state what result answers what part of the question. Answer exactly what is asked; you do not get any credit for making up your own questions and answering those. Use the stated procedures. Give exact, fully simplified, answers where possible.
One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a calculator, and a handwritten letter-size formula sheet.
Question: Analyze and very neatly graph
Question: Derive the dimensions (height and bottom radius) of
a conical tent of a volume at least equal to that requires the
least cloth area (in the conical surface). What is the ratio of
height over radius?
Question: In polar coordinates , consider the
region
inside the right half plane
that has boundary
. Sketch this region in the
-plane. (Plot
versus
before doing this and note that only
values with
are possible.) Now find the area of this
region, computing the double integral both doing
first and doing
first. Of course you should get the same value in either
case. Which method is easier? Note: integrate the region as given;
do not integrate half of it and multiply by 2.