Hello I need help with my calculus 3 assignment tomorrow from 7am-9am
Directions: Do not simplify unless indicated. No calculators are permitted. Show all work as appropriate for the methods taught in this course. Partial credit will be given for any work, words or ideas which are relevant to the problem. Important: Out of all the integrals that ask to be evaluated only one of them actually requires integration, the rest use theorems which make them fast. 1. Consider the following three pictures of vector fields. Each axis goes from −3 to 3 in each of [20 pts] the x and y directions and the vectors are anchored at integer coordinates. (I) (II) (III) For each of the following (a), (b), and (c), plug in a single point (x, y) which allows you to identify which of the vector fields (I), (II), or (III) matches the vector field. (a) F(x, y) = 0.3y i− 0.3x j (b) F(x, y) = 0.3y i + 0.3x j (c) F(x, y) = 0.3x i + 0.3 j 2. Parts (a) and (b) are unrelated. (a) Let Σ be the sphere with equation x2 + y2 + z2 = 9 with inwards orientation. Evaluate: [8 pts]∫∫ Σ (x i + 2y j + 3z k) · n dS (b) Suppose C is the curve parametrized by r(t) = t2 i + t j + 2 k for 0 ≤ t ≤ 2. Evaluate: [12 pts]∫ C y dx + z dy + z dz 3. Suppose Σ is the portion of the parabolic sheet y = 4−x2 in the first octant and below z = 3. [20 pts] The mass density at each point is given by f(x, y, z) = x + y + z. Set up an iterated integral for the mass of Σ. Do Not Evaluate This Integral. 4. Parts (a) and (b) are unrelated. (a) Let C be the clockwise triangle with vertices (5, 0), (5, 10), and (0, 10). Evaluate: [10 pts]∫ C (x + 2y) dx + (4x + y) dy (b) Let C be the straight-line segment from (1, 2,−2) to (8, 0, 5). Evaluate: [10 pts]∫ C (y i + x j + 1 k) · dr 5. Suppose C is the intersection of the plane 2x + y = 10 with the cylinder x2 + z2 = 9 with [20 pts] counterclockwise orientation when viewed towards the origin from the positive y-axis. Apply Stokes’ Theorem to the integral∫ C (x + y) dx + y2 dy + (xyz) dz and proceed until you have an iterated double integral. Do Not Evaluate This Integral. 1. Show all work 1. Consider the following three items: L: The line with symmetric equation x−1 2 = 3− y = z 8 P : The plane with equation x− 2y + 5z = 61 S: The sphere with equation x2 + (y − 2)2 + (z + 1)2 = 1 (a) Determine where L meets P . (b) It turns out that L does not meet S. You do not need to prove this. How close is L to the surface of S? 2. Consider the two curves C1 and C2 parameterized by: C1: r̄1(t) = t 2 ı̂ + (t− 2) ̂ + 5 k̂ C2: r̄2(t) = e t−1 ı̂ + cos(πt) ̂ + 5t k̂ (a) Show that the curves meet at t = 1 and do not meet at any other t-values. (b) Find the cosine of the angle between their velocity vectors at t = 1. (c) Which of them curves is longer between t = 1 and t = 2? Note: There are two integrals involved and you need to explain why one is larger than the other. You don’t need to actually integrate either, however. 3. Suppose C is the closed curve parameterized by: r̄(t) = (t− t2) ı̂ + (t2 − t3) ̂ for 0 ≤ t ≤ 1 This curve looks like this and is drawn counterclockwise: 0.1 0.2 0.3 0.1 0.2 Green’s Theorem tells us that for a region R with counterclockwise edge C we have∫ C (M ı̂ +N ̂ ) · dr̄ = ∫∫ R Nx −My dA. Thus specifically we have:∫ C (0 ı̂ + x ̂ ) · dr̄ = ∫∫ R 1 dA = Area of R Use this line integral to calculate the area of the region R inside the curve. 4. Suppose C is parameterized by: r̄(t) = 4 cos t ı̂ + (6− 4 sin t) ̂ + 4 sin t k̂ for 0 ≤ t ≤ 2π Using an appropriate surface apply Stokes’ theorem to the following integral. Proceed until you have an iterated double integral and then stop. Do Not Evaluate Your Final Iterated Integral! ∫ C (xy ı̂ − z2 ̂ + xz k̂ ) · dr̄ 5. Let C be the curve with parameterization: r̄(t) = 12t ı̂ + cos (2πt) ̂ + sin (2πt) k̂ for 0 ≤ t ≤ 1 12 The following line integral can be done two ways that we know. Show that both ways yield the same result. ∫ C y dx+ x dy + 3 dz 6. Instruction: Let A be 56 Let B be 9 Define the function: f(x, y) = Ax2y + 2ABxy + By2 (a) Let C be the level curve for f(x, y) = B. Determine where C meets the line y = 1. (b) Find and categorize each of the critical point of f as relative maximum, relative minimum, or saddle point. You should have three such points. 7. Instruction: Let E be 9 Let F be 1 Let D be the solid object in the first octant and bounded by the surfaces: F x + y = F and z = √ y. Let Σ be the surface of D. Suppose Σ is immersed in a fluid with flow F̄ (x, y, z) = x ı̂ + 2E 3 y3/2 ̂ − z k̂ . Find the rate at which F̄ is flowing inwards through Σ. 8. Instruction: Let G be 10 Let H be 21 Let R be the region in the first quadrant bounded by the four curves: y = Gx2 y = G x y = Hx2 y = H x Use the change of variables: u = y x2 and v = xy to evaluate the integral: ∫∫ R xy dA