4. Evaluate Sl y(+12) ds, where S = {(1,y,z) : 2 = 4 – r?,05152,0 <ys...
(,y,) dS, where f(,y,) = z'yz2 and S is the part ys 4 1. Parametrize, but do not evaluate, +y of the graph of z over the rectangle -2 S rs3 and 0 2. Parametrize, but do not evaluate, F.n dS, where F (y,-r,z) and S is the sphere of radius 2 centered at the origin. Math 224 3. Calculate le ayz dS where S is the part of the cone parametrized by 0sus1,0svs r(u, v)(ucos v, usin v, u),...
1. Find 12 + y² + 22 ds where is the helix r(t) = (a cost, a sint, bt) and 0 <t<l. 2. Evaluate |(2.84 +248, +16) - dr where C' is a curve that begins at (0,1) and ends at (1,2).
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...
A B C Parametrize, but do not evaluate, //f(x, y, z) ds, where f(x, y, z) 2y22 and S is the part , where J(,y,) 3 3 and 0 Sys4 of the graph of z2 over the rectangle -2 s . Parametrize, but do not evaluate, F.n ds, where F (,-,z) and S is the sphere of radius 2 centered at the origin. Calculate JJs xyz dS where S is the part of the cone parametrized by r(u, u) (ucos...
e.g.4 Evaluate JJs F dS, where j + sin(zy)k and S is the surface of the region E bounded by the parabolic cylinder z- 1 a2 and the planes z-0,y-0, and y + z-2. e.g.4 Evaluate JJs F dS, where j + sin(zy)k and S is the surface of the region E bounded by the parabolic cylinder z- 1 a2 and the planes z-0,y-0, and y + z-2.
Evaluate the surface integral lis(r,y,z) (x, y, z) ds where f(x, y, z) = x + y + z and o is the is the surface of the cube defined by the inequalities 0 < x < 5,0 Sy < 5 and 0 <3 < 5. [Hint: integrate over each face separately.] 1 f(x, y, z) ds =
Evaluate z) ds, where S is the intersection of the plane z=4-y with the solid cylinder x2 + y2 33. 8. 127211 ob.8V21 C. None of these O d. 4√3 a e. 1231
Evaluate the following: where S-( (z, y) є R2 : 0 ST/2,0 < y ST/2). (a) Jls (cosz-s (b) fdl where y is the line segment from (2,-1,3) to (0, 1, 4) and f (x,y,z)-y+2 sin y) dA 3 marks 3 marks (c) Jc F dr where C is the unit circle centred at the origin, traversed once anticlockwise and F R2R2 is given by F(r,y)- (x2.x + y) 3 marks JJR eVEdA where R is the region enclosed by...
S 6 Calculate the fleux SSF. ds, where I la, y, z)= <0? 2², 227 and S is the finite cylinder (with top and bottom) given by x² + y² = 1, 2 = 0, Z = 3,
Evaluate Z Z S curl(F) · dS where F(x, y, z) = (x^ 3 , −z ^3y ^3 , 2x − 4y) and S is the portion of the paraboloid z = x ^2 + y^ 2 − 3 below the plane z = 1 with orientation in the negative z-axis direction.