Math 53 Quiz 6
Problem 1 [5 pts]. Evaluate
Name:
1 1
0
y
x3 + 1 dxdy by changing the order of integration.
Problem 2 [5 pts]. Find the volume of the solid bounded by x + y = 1 and z = y in the rst octant. (Hint:
draw what is going on in the xy-plane)
1
Pr

Math 53 Quiz 7
Name:
Problem 1 [5 pts]. Evaluate the integral
y = 0, and y = 1.
E
x dV where E is the solid enclosed by the curves z = 1 x2 , z = 0,
Problem 2 [5 pts]. Using integration, nd the volume of the hemisphere (half-sphere) enclosed inside the un

MATH 53 SECTIONS 101 & 104 - QUIZ 7
Problem 1. (3 points) Find the volume of the wedge cut from the cylinder x2 + y2 = 1 by the
planes z = y and z = 0.
1
MATH 53 SECTIONS 101 & 104 - QUIZ 7
Problem 2.
2
(3 points) Find the volume of the region beneath the

Math 53 Quiz 5
Name:
Problem 1 [5 pts]. Consider the function xz + y ln z = 1. Dierentiate implicitly to nd
z
y .
Problem 2 [5 pts]. Find the equation for the plane that is tangent to the ellipse x2 + 2y 2 + 3z 2 = 4 at the point
(1, 0, 1).
1
Problem 3
(a

Math 53 Quiz 5, February 26, 2014
Maximum Points : 20 points, Time Allotted : 20 min
1. Make sure you show your work. Answers without justication will get no
points.
2. Box your nal answers.
3. Cross out anything you dont want me to consider.
4. There wil

Math 53 Quiz 6,March 12, 2014
Maximum Points : 20 points, Time Allotted : 20 min
1. Make sure you show your work. Answers without justication will get no
points.
2. Box your nal answers.
3. Cross out anything you dont want me to consider.
4. There will be

MATH 53 SECTIONS 101 & 104 - QUIZ 6
Problem 1. (4 points) Using Lagrange multipliers, nd the dimensions of the circular cylinder
with maximum volume that can be inscribed in a sphere of radius 16.
1
MATH 53 SECTIONS 101 & 104 - QUIZ 6
2
Problem 2. (3 poin

53 sections 102/103 Quiz 6
Name:
SID:
Problem 1 3 pts. Find the surface area of the paraboloid z = 5 x2 y 2 which lies in the rst
octant (x, y, z 0).
Problem 2 3 pts. Switch the order of integration to dy dz dx:
you will need two integrals.
1
1
0
1
0
1z
z

53 sections 102/103 Quiz 6
Name:
SID:
Problem 1 5 pts.
(a) 2 pts. Let g(x, y) = 3x2 + y 2 . Sketch (direction only; magnitude doesnt matter)
g(0, 6), and g(1, 3). Hint: think about what level set of g these points are on.
g( 2, 0),
(b) 3 pts. Find the max

53 sections 102/103 Quiz 5
Name:
SID:
Problem 1 1 pt/ea. True or false? Assume f in all problems is continuous, with all its partial
derivatives also continuous.
(a). T or F: If f is a function from R3 to R (with 3 inputs and one output) then rf is a func