MATH 127 Calculus III
Practice Problems for material since Midterm 2
The final exam will cover material from Sections 11.3, 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 12.7, 13.1, 13.2, 13.3, 14.1, 14.2,
14.3, 14.4, 14.5, 14.6, 14.7, 14.8, 15.1, 15.2, 15.3, 15.4, 15.5, 15.6, 16.1, 16.2, 16.3, 16.4, 16.5, 17.1, 17.2, 17.3.
Here is a list of some practice problems from the material we have covered that was not on the previous exams. This
should be used ONLY as a guide to the topics since midterm 2 and as problems for you to work. It should not be
seen as a comprehensive list. You may also find the practice exams for midterm 1 and 2, quizzes, LGAs, Achieve
problems and lecture slides also useful. It is important to understand all aspects of the course since the exam
is comprehensive/cumulative.
~ Find and sketch F.
~
(1) Suppose that f (x, y) = x2 − y is a potential function for F.
~
(2) Calculate the curl and divergence of the following vector field F(x,
y, z) = h0, cos(xz), − sin(xy)i.
(3) ˆ
Suppose f (x, y, z) = xyz and C is parametrized by ~r(t) = h2 sin(t), t, −2 cos(t)i for 0 ≤ t ≤ π. Compute
f (x, y, z) ds.
C
ˆ
f (x, y, z) ds where f (x, y, z) = x2 + y − z
(4) Let C be the line segment from (1, 1, 0) to (1, 1, 1). Evaluate
C
~
(5) Evaluate the vector line integral of F(x,
y, z) = hx + y, y − z, z 2 i over ~r(t) = ht2 , t3 , t2 i with 0 ≤ t ≤ 1.
~
(6) Calculate the work done by the vector field F(x,
y, z) = h−y sin(z), x sin(z), xy cos(z)i in moving a particle
2
2
2
around the circle cut from x + y + z = 10 by z = −1, clockwise as viewed from above.
~
(7) Consider the vector field F(x,
y, z) = hcos(z), −1, −x sin(z)i
~ conservative?
(a) Is F
~
(b) Find a scalar
ˆ potential function for F.
~ · d~r along C given by ~r(t) = het , e2t , ti from the point (1, 1, 0) to (eπ , e2π , π).
(c) Evaluate
F
C
√
(8) A particle starts at the point (−2, 0), and moves along the x-axis to (2, 0), and along the semicircle y = 4 − x2
~
to the starting point. Calculate the work done by the vector field F(x,
y) = hx, x3 +3xy 2 i in moving the particle
along the path.
˛
~ · d~r, where C = ∂D, if curlz (F)
~ = 6 in the region defined by the three curves and
F
(9) Find
C1
~ · d~r = π
F
~ · d~r = 7
F
~ · d~r = 3
F
C2
˛
˛
˛
C3
C4
2
~
(10) Compute the flux of the vector field F(x,
y, z) = hz, y, xi through the unit sphere x2 + y 2 + z 2 = 1 oriented
outward.
~
(11) Compute the flux of F(x,
y, z) = hx, y, 0i through the surface of the cylinder x2 + y 2 = 16 where 0 ≤ z ≤ 3,
oriented outwards.
(12) Consider the rectangular region S given below.
−−→
−−→
−−→
~
(a) Parametrize the region using the formula G(u,
v) = OA + uAB + v AD, with 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1.
~ u, G
~ v and N
~ =G
~u×G
~ v . Does N
~ point in the same direction as ~n?
(b) Compute G
¨
~
~ · d~S.
(c) Let F(x,
y, z) = h3x, 4y, 5i. Compute the vector surface integral
F
S
(13) For all parts of this questions, consider the surface S formed by the plane z = 7 − x and the solid cylinder
x2 + y 2 ≤ 16. The orientation is given in the figure below.
(a) Find a parametrization for the surface. ¨
~ · d~S for the vector field F
~ = hx2 , 0, y 2 i, where the
(b) Compute the vector surface integral
Curl(F)
orientation is given in the figure below.
~
(c) Compute the circulation of F.
S
~
(14) Find the flux of F(x,
yz) = hx2 y, xy 2 , 2xyzi outward through the sufrace of the solid bounded by the paraboloid
2
2
z = x + y and the plane z = 4.