integral problem

Pace  University   MAT  236  CRN  22205   Prof.  Kazlow   Spring  2015   Take-­‐home  Quiz  3   March  25,  2015

Exam  Instructions:    This  quiz  is  Due  Monday  May  5  in  class.  If  you  cannot  be  in   class  for  any  reason,  please  scan  and  submit  your  quiz  to  me  by  mail  or  submit  via   Blackboard  before  the  beginning  of  class.  You  may  use  your  book,  notes  and  the   internet  to  study  the  material  covered  by  the  exam.  You  may  not  use  internet   sources,  CAS  like  Maple,  Mathematica,  nor  Sage  to  supply  you  with  the  answers.     Please  submit  your  work  together  with  the  quiz.  Only  your  quiz  answers  will  be   viewed.  Your  work  will  be  checked  only  to  verify  that  you  did  the  work.  

Point  values:    10  points  each.  Maximum  Total  is  120  points.    

1. Find  the  local  maximum  and  minimum  values  and  note  where  they  occur.  Locate   all  saddle  points:    �  

2. Find  the  absolute/global  maximum  and  minimum  values  of  f  on  the  set  D.  Note   the  points  in  the  domain  at  which  they  occur:  � on   �  

3. Use  Lagrange  multipliers  to  Yind  the  maximum  and  minimum  values  of  the   function�  subject  to  the  constraint  �  

4. Consider:  �  deYined  by�  and  �   deYined  by  � .  Use  the  chain  rule� e  to  

Yind  the  derivative  of  � .     5. Evaluate  � ,  where  �  

6. Express  the  integral  � in  the  reverse  order  (i.e.,  

� .  

7. Find  the  volume  of  the  solid  bounded  by  the  plane  � and  the  paraboloid   �  by  evaluating  the  integral    where  � .  

� 


f (x,y) = x2 + xy + y2

g(x,y) = 4x + 6y − x2 − y2

D = {(x,y) |0 ≤ x ≤ 4,0 ≤ y ≤ 5}

f (x,y) = x2y x2 + 2y2 = 6 f :!3 →! f (x,y,z) = x3 + 3xyz − y2z g :!2 →!3

g(s,t) = 2t + s,−t − s,t2 + s2 1− x2 − y2( )dA D ∫∫

D( f !g)(1,2) y2 dA

D ∫∫ D = {(x,y) |−1≤ y ≤1,−2− y ≤ x ≤ y}

f (x,y)dy x2+1

5

∫⎡⎣⎢ ⎤ ⎦⎥ dx

1

2

∫ f (x,y)dx

a

b

∫⎡⎣⎢ ⎤ ⎦⎥ dy

c

d

∫ z = 0

z =1− x2 − y2 D = {(x,y) | x2 + y2 ≤1}

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Pace  University   MAT  236  CRN  22205   Prof.  Kazlow   Spring  2015   Take-­‐home  Quiz  3   March  25,  2015

9. Find  the  surface  area  of  the  part  of  the  paraboloid  �  that  lies  within  the   cylinder  � .  

�   10.  Find  the  limits  of  integration  for  calculating  the  volume  of  the  solid  Q  enclosed  

by  the  graphs  �  if  �  is  computed  by  repeated  

integration.  

�   11. Find  � where  R  is  the  parallelogram  with  vertices  

� .  

12. Evaluate  �

x = y2 + z2

y2 + z2 = 4

y = x2,z = 0,y + z = 2 Volume = 1dV Q ∫∫∫

(4x + 8y) R ∫∫ dA

(−1,−1),(1,3),(3,1),(5,5)

cos(x + y + z)dz 0

x

∫ dx0 y

∫ dy0 π /2

∫

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