Final exam, Math 240: Calculus III
April 29, 2005
No books, calculators or papers may be used, other than
a hand-written note card at most 5′′ × 7′′ in size.
For this web version, answers are at the end of the exam.
This examination consists of eight (8) long-answer questions and four (4) multiple-choice questions.
Each problem is worth ten points. Partial credits will be given only for long-answer questions,
when a substantial part of a problem has been worked out. Merely displaying some formulas is not
sufficient ground for receiving partial credits.
• Your name, printed:
• Your Penn ID (last 4 of the middle 8 digits):
• Your signature:
• Your lecture section (circle one):
Chai Caldararu
1 2 3 4 5 6 7 8 9-12 Total
, Part I. Long-answer Questions.
1. Compute det(A3 ), where A is the matrix
1 2 3
A= 1 4 9 .
1 8 27
2. Let C be the oriented curve
C = (x, y) : 4x2 + 9y 2 = 36, x ≥ 0, y ≥ 0
from (3, 0) to (0, 2). Compute the line integral
Z
(x + 1) dy + y dx .
C
3. Let D be the cube
D = (x, y, z) ∈ R3 0 ≤ x, y, z ≤ 1 ,
and let S = ∂D be the boundary surface of D, oriented by the unit normal vector field ~n on
S pointing away from D. Compute the oriented surface integral
ZZ
(x2 ~i + xyz ~j + z 3 ~k) · ~n dS .
S
4. Let S be the surface
S = (x, y, z) ∈ R3 x2 + y 2 + z 2 = 1, z ≥ 0 ,
the upper half of the unit sphere centered at the origin, oriented by the unit normal vector
field ~n = x~i + y ~j + z ~k on S. Compute the surface integral
ZZ
(x~i − y ~j + z ~k) · ~n dS .
S
5. Let C be the boundary of the rectangle with vertices (3, 2), (−5, 2), (−5, −7) and (3, −7),
oriented counter-clockwise. Compute the line integral
y dx − x dy
I
.
C x2 + y 2
1
April 29, 2005
No books, calculators or papers may be used, other than
a hand-written note card at most 5′′ × 7′′ in size.
For this web version, answers are at the end of the exam.
This examination consists of eight (8) long-answer questions and four (4) multiple-choice questions.
Each problem is worth ten points. Partial credits will be given only for long-answer questions,
when a substantial part of a problem has been worked out. Merely displaying some formulas is not
sufficient ground for receiving partial credits.
• Your name, printed:
• Your Penn ID (last 4 of the middle 8 digits):
• Your signature:
• Your lecture section (circle one):
Chai Caldararu
1 2 3 4 5 6 7 8 9-12 Total
, Part I. Long-answer Questions.
1. Compute det(A3 ), where A is the matrix
1 2 3
A= 1 4 9 .
1 8 27
2. Let C be the oriented curve
C = (x, y) : 4x2 + 9y 2 = 36, x ≥ 0, y ≥ 0
from (3, 0) to (0, 2). Compute the line integral
Z
(x + 1) dy + y dx .
C
3. Let D be the cube
D = (x, y, z) ∈ R3 0 ≤ x, y, z ≤ 1 ,
and let S = ∂D be the boundary surface of D, oriented by the unit normal vector field ~n on
S pointing away from D. Compute the oriented surface integral
ZZ
(x2 ~i + xyz ~j + z 3 ~k) · ~n dS .
S
4. Let S be the surface
S = (x, y, z) ∈ R3 x2 + y 2 + z 2 = 1, z ≥ 0 ,
the upper half of the unit sphere centered at the origin, oriented by the unit normal vector
field ~n = x~i + y ~j + z ~k on S. Compute the surface integral
ZZ
(x~i − y ~j + z ~k) · ~n dS .
S
5. Let C be the boundary of the rectangle with vertices (3, 2), (−5, 2), (−5, −7) and (3, −7),
oriented counter-clockwise. Compute the line integral
y dx − x dy
I
.
C x2 + y 2
1
