5. (25 points) Let S be the union of the following: • The portion of the...
Let S be the union of the following: • The portion of the cylinder x ^2 + y ^2 = 4 where x ≥ 0, bounded between the planes z = 0 and z = 2. • The rectangle −2 ≤ y ≤ 2, 0 ≤ z ≤ 2 in the yz-plane. Evaluate the integral Z Z S xz dS
Let S be the union of the following: • The portion of the cylinder x ^2 + y ^2 = 4 where x ≥ 0, bounded between the planes z = 0 and z = 2. • The rectangle −2 ≤ y ≤ 2, 0 ≤ z ≤ 2 in the yz-plane. Evaluate the integral Z Z S xz dS
Let Surface S be that portion of the cylinder x2 + y2 = 9, which lies between the planes z = y and z = 6. a.) Sketch the Surface S. b.) Parametrize the Surface S. c.) Evaluate the following Surface Integral: ∫∫(y-z)dS
Evaluate the integral. 3. Sss (xz – yz)ds; where S is portion of the plane in R3 z = x + y + 2, that lies inside the cylinder x2 + y2 = 1.
Could you do number 4 please. Thanks 1-8 Evaluate the surface integral s. f(x, y, z) ds Vx2ty2 -vr+) 1. f(x, y, z) Z2; ơ is the portion of the cone z between the planes z 1 and z 2 1 2. f(x, y, z) xy; ơ is the portion of the plane x + y + z lying in the first octant. 3. f(x, y, z) x2y; a is the portion of the cylinder x2z2 1 between the planes...
Please solve with detailed expplanation and graphs. Thank you! 8. Set up the following integrals in whatever coordinate system is most appropriate; use symmetry to simplify the integral if possible. You do not need to evaluate the integrals. ry+xz+yz) dV, where A is the region bounded by + y2 = 16 and the planes 2 = 0 and 2 = 4-y. -2 +32°) dV, where B is the region bounded by y = 4 - x, and the planes y...
Q3(a) Let W be the region above the sphere x2 + y2 + z2 = 6 and below the paraboloid z = 4 - x2 - y2 as shown in Figure Q5(a) below: Z=4-x-y? x2 + y + z = 6 Figure Q3(a) (i) Find the equation of the projection of Won the xy-plane. (ii) Compute the volume of W using polar coordinates. [16 marks] (b) Using double integral in polar coordinates, compute the following: $$*** (2x+3y) dedy [7 marks]...
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly, b) using Green's Theorem, where C consists of the arc of the parabola y = x2 from (0,0) to (2,4) and the line segments from (2,4) to (0,4) and from (0,4) to (0,0). [Answer: 0] Q2. Use Green's Theorem to evaluate the line integral $. F. dr or the work done by the force field F(x, y) = (3y - 4x)i +(4x - y)j...
5. Calculate the surface area of the portion of the sphere x2+y2+12-4 between the planes z-1 and z ะไ 6. Evaluate (xyz) dS, where S is the portion of the plane 2x+2y+z-2 that lies in the first octant. 7. Evaluate F. ds. a) F = yli + xzj-k through the cone z = VF+ア0s z 4 with normal pointing away from the z-axis. b) F-yi+xj+ek where S is the portion of the cylinder+y9, 0szs3, 0s r and O s y...
QUESTION 5 Let the surface S be the portion of the cylinder x2 + y2 4 under z 3 and above the xy-plane Write the parametric representation r(z,0) for the cylinder x2 +y2 4 in term of z (a) and 0 (2 marks) Based on (a), find the magnitude of llr, x rell for the given cylinder (b) (6 marks) 1 1+ (e) Evaluate z dS for the given S (8 marks) Hence, use the divergence theorem to evaluate f,...