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Math 114 Practice Problems for Test 3
Comments:
- Surface integrals, Stokes’ Theorem and Gauss’ Theorem used to be in the Math syllabus until last year, so we will look at some of the questions from those old exams for practice. Some of these will be challenging, but do not get discouraged. Ask me for hints if necessary.
- Your Test will be a mix of multiple choice questions and open answer questions.
- Short solutions are provided at the end.
Problem 1: (Spring 2009) Evaluate the integral
R x^ +^ y^ dA^ where^ R^ is the region inside the triangle with vertices (x, y) = (0, 0), (2, 0), and (0, 1).
(a) 0 (b) 1/ 4 (c) 1/ 3 (d) 1/ 2 (e) 2/ 3 (f ) 3/ 4 (g) 1 (h) 4/ 3
Problem 2: (Spring 2009) A cylinder of solid metal is given by the region in space bounded by x^2 + y^2 = 25 and the planes z = 0 and z = 4. The density function of the cyliner is ρ(x, y, z) = ex (^2) +y 2
. What is the mass of the cylinder?
(a) 4π(e^25 − 1) (b) 8π (c) 8π(e^5 − 1) (d) 10π (e) 10πe^16 (f ) π(e^10 − 1) (g) π(e−^25 − 1)/ 4 (h) 0
Problem 3: (Fall 2009) Let R be the region in the plane bounded by the square with vertices (0, 1), (1, 2), (2, 1), and (1, 0). Evaluate the integral ∫ ∫
R
(x + y)^2 sin(x − y)dA.
Problem 4: (Spring 2010) Compute the integral
∫ ∫
R
cos
x − y x + y
dA
where R is the region inside the triangle with vertices (0, 0), (0, 1), and (1, 0).
(a) −
2 (b) π/ 4 (c) 1 − cos 1 (d) 1 − √^12 (e) π/ 2 (f ) sin 1 2 (g) 0 (h) sin 1 − cos 1
Problem 5: (Math240 Spring 2007) Let C be the arc of the parabola x = t, y = 2−t−t^2 given by − 2 ≤ t ≤ 1 , and F = 2xex (^2) − 1 cos(y)i − ex (^2) − 1 sin(y)j be a vector field. Evaluate the line integral (^) ∫
C
F · dr.
(a) 12 (e + e−^1 ) (b) 12 (e^2 + e−^2 ) (c) e^3 − e−^1 (d) 12 (e − e−^1 ) (e) 1 (f ) 0 (g) 1 − e^3 (h) e^3 − 1
Problem 6: (Math240 Spring 2007) Let C denote the circumference (x−2)^2 +(y−2)^2 = 1 traversed counterclockwise. Evaluate the line integral ∮ (x^6 + 3y) dx + (2x − ey
2 ) dy.
(a) 0 (b) e^4 (c) − e^4 (d) − π (e) − 2 π (f ) 2π (g) π (h) π − e^4
Problem 7: (Math240 Spring 2007) Let S be the portion of the cone z = 1 −
x^2 + y^2 lying about the xy-plane. We orient S by a unit upward normal n. Given a vector field F = yi + sin(z^2 )j + cos(x^2 )k, evaluate the surface integral
∫ ∫
S
curl(F) · n dσ.
(a) sin(π^2 ) (b) π (c) − π (d) − 2 π (e) cos(π^2 ) − sin(π^2 ) (f ) 2π (g) − sin(π^2 ) (h) 0
Problem 8: (Math240 Spring 2007) Let S be the sphere x^2 + y^2 + z^2 = 4 oriented by the outward unit normal n = 12 (xi + yj + zk) and
F = (xy + x)i + (y − y^2 )j + (yz + z)k
be a vector field. Evaluate the surface integral
∫ ∫
S
F · n dσ.
(a) − 32 π (b) 32π (c) − 8 π (d) 16π (e) 163 π (f ) 8π (g) − 16 π (h) 0
Problem 9: (Math240 Fall 2007) Evaluate the line integral
C F^ ·^ dr^ in which^ C^ is the curve r(t) = 〈t, t^2 , t^3 〉 for 0 ≤ t ≤ 1 and F is the vector field 〈ey, xey, (z + 1)ez^ 〉.
(a) 12 (b) 2e (c) 0 (d) e (e) e 2
Problem 10: (Math240 Fall 2007) Evaluate the line integral
∫
C
(y + e
√x ) dx + (2x − cos(y^2 )) dy,
continuous derivatives. For each problem, state whether the given identity is true or false. You do not need to show any work.
(a) div(∇f ) = 0 True False (b) curl(∇f ) = 0 True False (c) div(curl F) = 0 True False (d) curl(curl F) = 0 True False (e) ∇(div F) = 0 True False
Problem 15: (Math240 Spring 2008) Define the function
f (x, y, z) = e(sin^ x^ cos^ y)
z +
π 2
Let C be the curve
r(t) = 〈t cos^2 (2t), t sin(t), t〉, for 0 ≤ t ≤ π/ 2.
Compute the integral (^) ∫
C
∂f ∂x
dx +
∂f ∂y
dy +
∂f ∂z
dz.
Problem 16: (Math240 Spring 2008) Let S be the closed surface in 3 -space formed by the cone x^2 + y^2 − z^2 = 0, 1 ≤ z ≤ 2 , the disk x^2 + y^2 ≤ 4 in the plane z = 2, and the disk x^2 + y^2 ≤ 1 in the plane z = 1. Define the vector field
F(x, y, z) = xy^2 i + x^2 yj + sin xk,
and let∫∫ n be the outward pointing unit normal vector S. Compute the surface integral
S F^ ·^ n^ dσ.
Problem 17: (Math240 Spring 2010) Find the work done by the force field
F(x, y, z) = eyi + (xey^ + ez^ )j + yez^ k
in moving a particle from (1, 0 , 0) to (0, 1 , π) along the helix x = cos t, y = sin t, z = t.
(a) eπ^ (b) eπ^ − 2 (c) eπ^ + 1 (d) eπ^ − 1 (e) 2eπ^ − 1 (f ) 2eπ^ − 3
Remark. This is a verbatim copy of the exam question. Can you spot an inconsistency in the question itself? How might you fix it?
Problem 18: (Math240 Spring 2010) Let C be the curve that is the intersection of the plane x + y + z = 1 and the cylinder x^2 + y^2 = 9, oriented counter-clockwise as viewed from above. Evaluate
C F^ ·^ dr^ where
F(x, y, z) = −yx^2 i + y^2 zj + z^2 k.
(a) 0 (b) 12 (c) 13 (d) − 2 (e) − (^14)
Problem 19: (Math240 Spring 2010) Let F(x, y) = 〈y^2 , 3 xy〉 be a vector field in the plane and let C be the closed curve consisting of four piecewise smooth pieces where C 1 is the top half of the circle x^2 + y^2 = 4, C 3 is the top half of the circle x^2 + y^2 = 1, and C 2 and C 4 are line segments of unit length along the x-axis which connect the two semicircles. Orient this curve in a counter-clockwise orientation. Evaluate the integral∮
C F^ ·^ dr. (a) − 10 (b) 143 (c) 103 (d) − 105 (e) (^32)
Problem 20: (Math240 Spring 2010) Find the outward flux
S F^ ·^ n^ dσ^ of the vector field F = 3xy^2 i + 3yz^2 j + 3zx^2 k where the surface S is the boundary of the region 1 ≤ x^2 + y^2 + z^2 ≤ 4.
Problem 21: (Math240 Spring 2010) Which of the following are true or false?
(a) If S is any closed surface, then
S ∇ ×^ F^ ·^ n^ dσ^ = 0.^ True^ False (b) If F = ∇f , then
C F^ ·^ dr^ = 0^ for all closed curves^ C.^ True^ False
Problem 22: (Math240 Spring 2009) The portion of the plane z = 10 + 2x + 3y over the disc x^2 + y^2 ≤ 1 has area equal to
(a) π/
14 (b)
14 (c) 10 +
14 (d)
14 π (e) π
14 (f ) 4π
13 (g) 4π/
14 (h) 14π
Problem 23: (Math240 Spring 2009) The flux of the field F = xi + yj − zk across the cylindrical surface x^2 + z^2 = 1, 0 ≤ y ≤ 3 with outward pointing normal equals:
(a) − 3 π (b) 0 (c) π (d) 2π (e) 3π (f ) 4π (g) 6π (h) 9π
Problem 24: (Math240 Spring 2009) Let S denote the sphere of radius r centered at the origin. If the flux of a vector field F across S with outward pointing normal is 8 πr^3 / 3 , which of the following could be F? Choose one.
(a) F = 2rxi + 2ryj + 2rzk (b) F = xi + 2yj + zk (c) F = xi + zj + yk (d) F = xi − xzj + zk (e) F = i + j + k (f ) F = 2xi + 2yj + 2zk (g) F = zi + xj + yk (h) F = 2i + 2j + 2k
Problem 25: (Math240 Spring 2009) The change of variables x = 2u, y = v − u^2 transforms the integral
∫ (^2)
0
∫ (^) u (^2) +
u^2
4 u sin(v − u^2 )e^2 uv−^2 u
3 dvdu
1: G
2: A
4: F
5: G
6: D
7: C
8: B
9: B
10: A
11: D
12: C
13: F
14: FTTFF
15: π/ 2
16: 31π/ 10
17: D, sort of
18: A
19: B
20: 372π/ 5
21: TT
22: E
23: B
24: D
25: F
26: G
