AMath 231 ASSIGNMENT # 1: Review and Curves Spring 2014

Due on or before Thursday, May 15th at noon in location to be announced. Late assign-ments or those put into the wrong drop slot will not be marked.

Comment: The first two problems are review. If you took Math 247 (Advanced Calculus3) then you might not have had sufficient practice in evaluating double or triple integrals.If this is the case, it is strongly recommended that you review the Math 237 course notes.Contact your instructor if you have difficulty obtaining these.

1. Evaluate

D

xy2 dA where D is the region bounded by y = x, y = 2x and x = 3.

2. Suppose that a hemispherical tank (with the flat part on the ground) with radius Ris partially filled with water, so that the depth of the water is h. Find the volumeof water in the tank. Consider spherical and cylindrical coordinates before trying toevaluate; one of them will be easier to use.

PS1 # 3, 5, 8, 11

Note: PS1 refers to problem set 1 in the back of the course notes. On your assignment,label your solution to question 3 with PS1 # 3, for example, instead of just 3, toavoid confusion with the other # 3.

3.* Can the Mean Value Theorem be generalized to vector-valued functions, i.e. is thefollowing statement true?

If g C1[a, b], then there exists a number c with a < c < b such that

g(b) g(a) = (b a)g(c).

*Questions marked with an asterisk (*) are considered challenging. Challenging questionswill also appear on the exams, but will normally be worth at most 5% of the grade and arebalanced by 10% easy marks.