Integral Theorems Review Solutions
Math 21a
1
Find the line integral
0 t 2 .
C
Fall, 2010
F dr, where F = 6z, y 2 , 12x , and C : r(t) = sin t, cos t, t/6 , with
Since the vector field isnt conservative and isnt too ugly, we evaluate the integral directly
Math 21a
1
Integrals in Cylindrical & Spherical Coordinates
Fall, 2009
Using cylindrical coordinates, evaluate the integral
x2 + y 2 dV , where E is the solid in
E
2
2
the first octant inside the cylinder x + y = 16 and below the plane z = 3.
3
/2
4
r2 d
Math 21a
1
Exam II Practice Questions (Solutions)
Fall, 2010
True/False:
(a) There exists a function f (x, y ) of two variables with two local maxima and no local
minima. True.
(b) If f (x, y ) is a continuous positive function on the region R = cfw_(x, y
Triple Integrals
Math 21a
1
Evaluate the integral
E
2x dV , where
E = cfw_(x, y, z ) : 0 y 2, 0 x
2
0
2
Fall, 2010
4 y 2
0
4 y 2 , 0 z y .
y
2x dz dy dx = 4.
0
Sketch the solid whose volume is given by the integral
1
0
1 x
0
2 2z
0
dy dz dx.
See problem
Math 21a
1
Double Integrals III: Polar Coordinates
Evaluate the integral
circle x2 + y 2 = 9.
R
Fall, 2010
cos(x2 + y 2 ) dA, where R is the region above the x-axis within the
In polar coordinates, we have
2
3
2
cos(x + y ) dA =
R
2
0
0
1
cos(r2 )r dr d =
Double Integrals II
Math 21a
1
Evaluate the double integral
and x = 1. (Draw the region!)
We set up the integral as
2
1
0
D
x2
0
x cos y dA, where D is the region bounded by y = 0, y = x2 ,
x cos y dy dx. Evaluate to get sin2 (1/2).
Sketch the region of i
Math 21a
1
Double Integrals
Fall, 2010
Let f (x, y ) = xy and let R = [0, 6] [0, 4].
(a) Subdivide R into a 3 2 grid of congruent squares, and let the six points ( xi , yj ) (1 i
3, 1 j 2) be the midpoints of these squares. (Draw a picture!) Use this sub
Math 21a
1
Global Min/Max
Fall, 2010
Find the extreme values of the function f (x, y ) = 2x3 + y 4 on the closed disk D of radius 1
centered at the origin.
The maximum value is 2, occurring at (1, 0); the minimum is 2, occurring at (1, 0). If this
isnt wh
Math 21a
1
Lagrange Multipliers
Fall, 2010
Use Lagrange multipliers to find the maximum and minimum values of the function f (x, y ) =
x2 + y 2 subject to the constraint xy = 1.
No maximum; minimum is 2, occurring at (1, 1) and (1, 1).
2
The Cobb-Douglas
Chain Rule & Implicit Differentiation
Math 21a
1
Use the chain rule to find dw/dt, where w = ln
z = tan t.
dw
dt
Fall, 2010
x2 + y 2 + z 2 , x = sin t, y = cos t, and
= tan t.
2
Use the chain rule to find the partial derivatives fs = f /s and ft = f /t, w
Math 21a
1
Tangent Planes & Linear Approximation
Fall, 2010
Find the tangent plane to the surface z = ln(x 2y ) at the point (3, 1, 0).
(x 3) 2(y 1) = z
2
Find an equation of the tangent plane to the parametric surface r(u, v ) = u2 , 2u sin v, u cos v
at
Math 21a
1
Midterm I practice questions (Solutions)
Fall, 2010
True/False:
(a) For any three points P, Q, R in space, P Q P R = QP P R. False.
(b) If |r (t)| is constant, then r (t) and r (t) are always orthogonal. True.
(c) The vectors 2, 2, 1 and 1, 1,
Math 21a
Cylindrical and Spherical Coordinates
Fall, 2009
3, 2 3) given in rectangular coordinates.
1
Find cylindrical coordinates for the point (1 ,
(2, /3, 2 3)
2
Identify the surface with equation z = 4 r2 in cylindrical coordinates.
The surface is a d
Quadric Surfaces
Math 21a
1
Fall, 2010
For each of the following surfaces, identify the x-traces, y -traces, and z -traces. Then give a
rough sketch of the surface.
(a)
z
x2 y 2
= 2 2
a
b
c
See the table on page 679 of the text.
(b)
x2 y 2 z 2
+ 2 2 =1
a2
Math 21a
1
Functions and Graphs
Fall, 2010
Draw a contour map for the function f (x, y ) = (y 2x)2 . What does the graph of f look like?
See Exercise 11.1.19, p. A41 in the text.
2
Consider the function f (x, y ) =
x + y.
(a) What is the domain and range