Math 53 Midterm #2, 4/10/07, 3:40 PM – 5:00 PM
(please do not leave the exam between 4:45 and 5:00)
No calculators or notes are permitted. Each of the 6 questions is worth
10 points. Please write your solution to each of the 6 questions on a separate
sheet of paper with your name, SID number, and GSI’s name on it. To get
full credit, you must put a box around your final answer and show correct
work/justification. Good luck!
1. Find the volume of the solid region between the surfaces z= 2x2+ 2y2
and z= 12 −x2−y2.
2. Find the minimum and maximum values of the function
f(x, y) = x2+y2+ 5y
on the region x2+y2≤4, and say where the function takes these values.
3. Evaluate the iterated integral
Z1
0Z1
x
cos y
ydy dx.
4. Evaluate the triple integral
ZZZE
(x2+y2+z2)3/2dV,
where Eis the region determined by the inequalities x2+y2+z2≤1,
z≥0, and z2≤x2+y2.
5. Let Rdenote the triangle in the x, y plane with corners at (0,0), (1,0),
and (0,1). Use the change of variables x=u2,y=v2to evaluate the
double integral
ZZR
1
√xy dA.
6. Evaluate the iterated integral
Z1/√2
0Z√1−x2
x
ex2+y2dy dx.