(a) 0 (b) 1/4 (c) 1/3 (d) 1/2 (e) 2/3 (f) 3/4 (g) 1 (h) 4/3

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1 Math 114 Practice Problems for Test 3 omments: 0. urface integrals, tokes Theorem and Gauss Theorem used to be in the Math40 syllabus until last year, so we will look at some of the questions from those old exams for practice. ome of these will be challenging, but do not get discouraged. Ask me for hints if necessary. 1. Your Test will be a mix of multiple choice questions and open answer questions.. hort solutions are provided at the end. Problem 1: (pring 009) Evaluate the integral x + y da where R is the region R inside the triangle with vertices (x, y) = (0, 0), (, 0), and (0, 1). (a) 0 (b) 1/4 (c) 1/3 (d) 1/ (e) /3 (f) 3/4 (g) 1 (h) 4/3 Problem : (pring 009) A cylinder of solid metal is given by the region in space bounded by x + y = 5 and the planes z = 0 and z = 4. The density function of the cyliner is ρ(x, y, z) = e x +y. What is the mass of the cylinder? (a) 4π(e 5 1) (b) 8π (c) 8π(e 5 1) (d) 10π (e) 10πe 16 (f) π(e 10 1) (g) π(e 5 1)/4 (h) 0 Problem 3: (Fall 009) Let R be the region in the plane bounded by the square with vertices (0, 1), (1, ), (, 1), and (1, 0). Evaluate the integral (x + y) sin(x y)da. R Problem 4: (pring 010) ompute the integral ( ) x y cos da x + y R where R is the region inside the triangle with vertices (0, 0), (0, 1), and (1, 0). (a) (b) π/4 (c) 1 cos 1 (d) 1 1 (e) π/ (f) sin 1 (g) 0 (h) sin 1 cos 1 Problem 5: (Math40 pring 007) Let be the arc of the parabola x = t, y = t t given by t 1, and F = xe x 1 cos(y)i e x 1 sin(y)j be a vector field. Evaluate the line integral F dr. (a) 1 (e + e 1 ) (b) 1 (e + e ) (c) e 3 e 1 (d) 1 (e e 1 ) (e) 1 (f) 0 (g) 1 e 3 (h) e 3 1

2 Problem 6: (Math40 pring 007) Let denote the circumference (x ) +(y ) = 1 traversed counterclockwise. Evaluate the line integral (x 6 + 3y) dx + (x e y ) dy. (a) 0 (b) e 4 (c) e 4 (d) π (e) π (f) π (g) π (h) π e 4 Problem 7: (Math40 pring 007) Let be the portion of the cone z = 1 x + y lying about the xy-plane. We orient by a unit upward normal n. Given a vector field F = yi + sin(z )j + cos(x )k, evaluate the surface integral curl(f) n dσ. (a) sin(π ) (b) π (c) π (d) π (e) cos(π ) sin(π ) (f) π (g) sin(π ) (h) 0 Problem 8: (Math40 pring 007) Let be the sphere x + y + z = 4 oriented by the outward unit normal n = 1 (xi + yj + zk) and F = (xy + x)i + (y y )j + (yz + z)k be a vector field. Evaluate the surface integral F n dσ. (a) 3π (b) 3π (c) 8π (d) 16π (e) 16π 3 (f) 8π (g) 16π (h) 0 Problem 9: (Math40 Fall 007) Evaluate the line integral F dr in which is the curve r(t) = t, t, t 3 for 0 t 1 and F is the vector field e y, xe y, (z + 1)e z. (a) 1 (b) e (c) 0 (d) e (e) e Problem 10: (Math40 Fall 007) Evaluate the line integral (y + e x ) dx + (x cos(y )) dy,

3 where is the boundary of the region enclosed between the curves y = x and x = y and is oriented counter-clockwise. (a) 1 3 (b) 3 (c) 0 (d) π 3 (e) π Problem 11: (Math40 Fall 007) onsider the surface formed by the upper half of the ellipsoid x + y + 6z = 1, and write for the circle x + y = 1 where cuts the xy-plane. We use the outer normal (upward pointing) to orient so that is traversed counter-clockwise in the xy-plane when viewed from above. Let F be the vector field sin(xz) + x, (3 + z)x e y, x y 3 z 5. ompute the surface integral F n dσ. (a) π (b) π (c) 3π (d) 3π (e) none of the above Problem 1: (Math40 Fall 007) Write for the part of the surface z = x +y over the disc x + y 1 in the plane. We orient so that its normal n points downward. If the vector field F is given by F(x, y, z) = e y + x, e sin(z) + sin(x), z + xy, then the surface integral F n dσ equals (a) π (b) 4π (c) π (d) π/ (e) none of the above Problem 13: (Math40 pring 008) onsider the vector field ) F = ( z 5 z + πyesin x cos x i + ( πe sin x x ) j xz 5 k and the curve given by Evaluate the line integral F dr. r(t) = cos t, sin t, 0, for π/ t π/. (a) π (b) 0 (c) 4π (d) π (e) π (f) π (g) none of the above Problem 14: (Math40 pring 008) In the following, F is any vector field in 3 dimensions and f is any function in 3 variables. You may assume F and f have

4 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: (Math40 pring 008) Define the function ( f(x, y, z) = e (sin x cos y) z + π ). Let be the curve r(t) = t cos (t), t sin(t), t, for 0 t π/. ompute the integral f f f dx + dy + x y z dz. Problem 16: (Math40 pring 008) Let be the closed surface in 3-space formed by the cone x + y z = 0, 1 z, the disk x + y 4 in the plane z =, and the disk x + y 1 in the plane z = 1. Define the vector field F(x, y, z) = xy i + x yj + sin xk, and let n be the outward pointing unit normal vector. ompute the surface integral F n dσ. Problem 17: (Math40 pring 010) Find the work done by the force field F(x, y, z) = e y i + (xe y + e z )j + ye z 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) e π (b) e π (c) e π + 1 (d) e π 1 (e) e π 1 (f) e π 3 Remark. This is a verbatim copy of the exam question. an you spot an inconsistency in the question itself? How might you fix it? Problem 18: (Math40 pring 010) Let be the curve that is the intersection of the plane x + y + z = 1 and the cylinder x + y = 9, oriented counter-clockwise as viewed from above. Evaluate F dr where F(x, y, z) = yx i + y zj + z k. (a) 0 (b) 1 (c) 1 3 (d) (e) 1 4

5 F dr. (a) 10 (b) 14 3 (c) 10 3 (d) 10 5 (e) 3 Problem 19: (Math40 pring 010) Let F(x, y) = y, 3xy be a vector field in the plane and let be the closed curve consisting of four piecewise smooth pieces where 1 is the top half of the circle x + y = 4, 3 is the top half of the circle x + y = 1, and and 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 Problem 0: (Math40 pring 010) Find the outward flux F n dσ of the vector field F = 3xy i + 3yz j + 3zx k where the surface is the boundary of the region 1 x + y + z 4. Problem 1: (Math40 pring 010) Which of the following are true or false? (a) If is any closed surface, then F n dσ = 0. True False (b) If F = f, then F dr = 0 for all closed curves. True False Problem : (Math40 pring 009) The portion of the plane z = 10 + x + 3y over the disc x + y 1 has area equal to (a) π/ 14 (b) 14 (c) (d) 14π (e) π 14 (f) 4π 13 (g) 4π/ 14 (h) 14π Problem 3: (Math40 pring 009) The flux of the field F = xi + yj zk across the cylindrical surface x + z = 1, 0 y 3 with outward pointing normal equals: (a) 3π (b) 0 (c) π (d) π (e) 3π (f) 4π (g) 6π (h) 9π Problem 4: (Math40 pring 009) Let denote the sphere of radius r centered at the origin. If the flux of a vector field F across with outward pointing normal is 8πr 3 /3, which of the following could be F? hoose one. (a) F = rxi + ryj + rzk (b) F = xi + yj + zk (c) F = xi + zj + yk (d) F = xi xzj + zk (e) F = i + j + k (f) F = xi + yj + zk (g) F = zi + xj + yk (h) F = i + j + k Problem 5: (Math40 pring 009) The change of variables x = u, y = v u transforms the integral u + 0 u 4u sin(v u )e uv u3 dvdu

6 into which integral with respect to (x, y) coordinates? hoose one. (a) (d) (g) x sin(y)e xy dxdy x sin(y)e xy dxdy x sin(y)e xy dxdy (b) (e) (h) x sin(y)e xy dydx x sin(y)e xy dydx x sin(y)e xy dxdy (c) (f) x sin(y)e xy dxdy x sin(y)e xy dydx Problem 6: (Math40 pring 009) The curl of F is F = i+j+3k. What is the work done by F along the oriented square path from (1, 1, 3) to (, 1, 3) to (, 0, 3) to (1, 0, 3) and back to (1, 1, 3)? (a) 6 (b) 3 (c) (d) 1 (e) 1 (f) (g) 3 (h) 6

7 1: G : A 3: 0 4: F 5: G 6: D 7: 8: B 9: B 10: A 11: D 1: 13: F 14: FTTFF 15: π/ 16: 31π/10 17: D, sort of 18: A 19: B 0: 37π/5 1: TT : E 3: B 4: D 5: F 6: G