MATH 232, Section 1, Fall 2014, Quiz 1.
Last 4 digits of PSU ID:
Name:
Let D be the domain in R2 bounded by y = x, y = 1, x = 0.
1. Draw the domain D.
1
D
1
ex/y dA as an iterated integral for both or
Homework 7
Math 232, Alena Erchenko
1. Sketch the vector fields
y x
F~ (x, y) = p
x2 + y 2
and
~
G(x,
y, z) = + k.
Solution. The plots are below.
(Note that the first one it is actually undefined at t
Homework 9
Math 232, Alena Erchenko
1. Compute the curl of the following two vector fields.
~ = xy + xz
G
+ yz k
F~ = x2 y + xy 2
Solution. We have
curl F~ = Qx Py = x2 + y 2
and
~ = hRy Qz , Pz Rx
Homework 2
Math 232, Alena Erchenko
1. Set up and compute the integral of xy in both cartesian and polar coordinates over the region
bound by the x-axis, x = 1, and y = x.
Solution. In cartesian coord
Homework 12
Math 232, Alena Erchenko
1. Let C be the closed curve obtained by traveling along the straight line segments from (1, 0, 0)
to (0, 1, 0) to (0, 0, 1) and back to (1, 0, 0). Verify Stokes t
Homework 3
Math 232, Alena Erchenko
1. Find the average value of the function f (x, y) = y sin(xy) inside the unit circle and unit square
0 x 1, 0 y 1.
Solutions. Observe that f (x, y) = f (x, y) so t
Homework 11
Math 232, Alena Erchenko
1. Verify the divergence theorem for the vector field F~ = hx2 , x2 y, x2 zi and the surface S which
is the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1,
Homework 4
Math 232, Alena Erchenko
1. Evaluate the integral
Z
0
1
Z
1x
x + y(y 2x)2 dy dx
0
by making the substitution. u = x + y and v = y 2x.
Solution. The Jacobian determinant is
1 1
2 1 = 3
so
d
Homework 8
Math 232, Alena Erchenko
R
1. Compute directly C F~ d~r where F~ = hy 2 + 2xy, x2 + 2xyi and C is the portion of the cusp
curve y 2 = x3 from the origin to the point (1, 1). Use the theorem
Homework 10
Math 232, Alena Erchenko
1. Use a surface integral to compute the surface area of cylinder (without top and bottom)
x2 + y 2 = a2 with radius a and height h.
Solution. Consider the base to
Homework 1
Math 232, Alena Erchenko
1. Evaluate the following double integrals:
R1Rx
(a) 0 x2 (1 + 2y) dydx
RR
(b) D x5y+1 dA, where D = cfw_(x, y)|0 6 x 6 1, 0 6 y 6 x2
RR
(c) D 2xy dA, where D is t
Homework 6 (optional)
Math 232, Alena Erchenko
As we saw before, there are a lot of similar formulas for double and triple integrals. In this
homework we will try to compute the volume of a n-dimensio
MATH 232, Section 1, Fall 2014, Quiz 6.
Name:
Last 4 digits of PSU ID:
F dS where F = xi + y j + z k and S is the
Question 1. Compute the surface integral
S
hemisphere x2 + y 2 + z 2 = 9, y 0, oriente
MATH 232, Section 1, Fall 2014, Quiz 1.
Last 4 digits of PSU ID:
Name:
Let D be the domain in R2 bounded by y = x, y = 1, x = 0.
1. Draw the domain D.
1
1
ex/y dA as an iterated integral for both orde
MATH 232, Section 1, Fall 2014, Quiz 2.
Name:
Last 4 digits of PSU ID:
Let E be the solid in R3 bounded by the sphere x2 + y 2 + z 2 = 1, the xy-plane, the xz-plane, and
such that y 0.
zdzdydx as an i
MATH 232, Section 1, Fall 2014, Quiz 2.
Last 4 digits of PSU ID:
Name:
Let E be the solid in R3 bounded by the sphere x2 + y 2 + z 2 = 1, the xy-plane, the xz-plane, and
such that y 0 and z 0.
zdzdydx
MATH 232, Section 1, Fall 2014, Quiz 4.
Last 4 digits of PSU ID:
Name:
Question 1. Evaluate the line integral
F dr where
C
2
1
F = ln(x + 3) y 2 , ey + 3xy
2
and C is curve shown in the Figure below c
MATH 230, Section 15, Fall 2014, Quiz 3.
Name:
Last 4 digits of PSU ID:
Question 1. Let S be the solid in the rst octant (i.e., the part of R3 such that x 0, y 0 and
z 0), above the cone of equation z
MATH 232, Section 1, Fall 2014, Quiz 4.
Last 4 digits of PSU ID:
Name:
Question 1. Evaluate the line integral
F dr where
C
2
1
F = ln(x + 3) y 2 , ey + 3xy
2
and C is curve shown in the Figure below c
MATH 230, Section 15, Fall 2014, Quiz 3.
Name:
Last 4 digits of PSU ID:
Question 1. Let S be the solid in the rst octant (i.e., the part of R3 such that x 0, y 0 and
z 0), above the cone of equation z
MATH 232, Section 1, Fall 2014, Quiz 5.
Name:
Last 4 digits of PSU ID:
Question 1. Evaluate the surface area of the cone S dened by z = x2 + y 2 , z 4. (Answers
that uses geometric formulas not seen i
MATH 232, Section 1, Fall 2014, Quiz 5.
Last 4 digits of PSU ID:
Name:
Question 1. Evaluate the surface area of the cone S dened by z = x2 + y 2 , z 4. (Answers
that uses geometric formulas not seen i
MATH 232, Section 1, Fall 2014, Quiz 6.
Name:
Last 4 digits of PSU ID:
F dS where F = xi + y j + z k and S is the
Question 1. Compute the surface integral
S
hemisphere x2 + y 2 + z 2 = 9, y 0, oriente
Homework 5 - Solutions
Math 232, Alena Erchenko
RRR
1. Evaluate
(x y) dV where E is the solid that lies between the cylinders x2 + y 2 = 1 and
E
2
2
x + y = 16, above the xy-plane, and below the plane