(sin(π/z) -1dd 2. Compute the integral: sin(π/s)-.--d γ is the cl γ is the curve shown in the 2. where 721-1 following figure: arked points on the coordinate axes correspond to T,-T, 2, 2. (sin(...
Question 11 3 pts Compute the line integral forma sin z ds, where is the curve in R3 with parametric equation r(t) = costi+ sintj+tk, 0 <t< /2. The value of the integral is So x2 sin z ds = 1
Problem 4.9 (e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
(3) For the following velocity fields F on R3, find the flow along the given curve. r(t) = (t, t2, 1) F=(-4xy, 83, 2) with 0 2 t 1l F=(z-z, 0,2) r(t)-(cost, 0, sin t) with 0 t π F = (-y,2, 2) with r(t) = (-2 cost, 2 sin t, 2t) 0 < t < 2π (3) For the following velocity fields F on R3, find the flow along the given curve. r(t) = (t, t2, 1) F=(-4xy, 83,...
Problem 3 (12 points) The curve with parametric equations (1 + 2 sin(9) cos(9), y-(1 + 2 sin(θ)) sin(0) is called a limacon and is shown in the figure below. -1 1. Find the point (x,y 2. Find the slope of the line that is tangent to the graph at θ-π/2. 3. Find the slope of the line that is tangent to the graph at (,y)-(1,0) ) that corresponds to θ-π/2. Problem 3 (12 points) The curve with parametric equations...
Question 3 [25 points]: Complex integration Subquestions (a), (b), and (c) will use C1 shown in the figure on the left-hand side, whereas subquestion (d) will use C2 shown in the figure on the right-hand side. Im (2) Im (2) SA= 1 → Re (2) → Re (2) 20 = 1 - (a) [3 points) Find a parametric representation for the curve Ci. (b) [7 points] Compute the integral Sc, z dz. (c) [5 points) Compute the integral Se, 22...
1. (2 points) Find F dF if curl(F) 3 in the region defined by the 4 curves and C4 Ci F . d7 where F(x,y,z)-Wi +pz? + Vi> and C consists of the arc of the 2. (2 points) Evaluate curve y = sin(x) from (0,0) to (π, 0) and the line segment from (π,0) to (0,0). 4 3 3. (2 points) Evaluate F di where F.y,(ry, 2:,3) and C is the curve of intersection of 5 and y29. going...
2) (15 points) (a) (10 points) Compute the line integral s f(x,y) ds of the scalar function over the oriented curve. [C] 0 (5 points) How does your answer change if I reverse the orientation of Cl? f(0, y) = C): The curve parameterized by r(t) = t'i + t'j, t E (1.21
Compute the Riemann sum S for the double integral (6x + 5y) dA where R = [1,4] × [1, 3], for the grid and sample points shown in figure below S = 155 Compute the Riemann sum S for the double integral (6x + 5y) dA where R = [1,4] × [1, 3], for the grid and sample points shown in figure below S = 155
Question 4 (10 points) Evaluate the line integral, where C is the given curve. y sin(2+1) dy where answer. consists of the line segment from (2,4.-1) to (1,-1,3). Select the correct where answer. consists of the line segment from (2,4,-1) to (1,-1,3). Select the correct 2) аn(a) - сот а) – 5 сов (1) - sin(41) - 5 ооооо V42 (4 cos(4) — 5 sin(4) + 16) None of these
(1 point) Let Vf =-8xe-r sin(5y) 20e-x. cos(Sy) j. Find the change inf between (0,0) and (1, π/2) in two ways vf . dr, where C is a curve connecting (0,0) and (1.d2). (a) First, find the change by computing the line integral The simplest curve is the line segment joining these points. Parameterize it: with 03t s 1, r(t)- so that Icvf . di- Note that this isn't a very pleasant integral to evaluate by hand (though we could...