Find the mass of the cylinder (centered on the z-axis and base in the xy-plane) with radius 4 and...
(1 point) Evaluate the triple integral of f(x, y, z) = cos(x2 + y2) over the solid cylinder with height 4 and with base of radius 2 centered on the z axis at z = -2. Integral
(1 point) Evaluate the triple integral of f(x, y, z) = cos(x2 + y²) over the solid cylinder with height 2 and with base of radius 1 centered on the z axis at z = -2. Integral = 6pisin(4)
(a) Set up a double integral for calculating the flux of the vector field F(x, y, z) = (x2, yz, zº) through the open-ended circular cylinder of radius 5 and height 4 with its base on the xy-plane and centered about the positive z-axis, oriented away from the z-axis. If necessary, enter 6 as theta. Flux = -MIT" dz de A= BE C= D= (b) Evaluate the integral. Flux = S]
Suppose E is the half-cylinder described by x^2 + y^2 = 1 between z = 4 and the xy-plane where y ≥ 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) Find an expression for the mass of E as a triple integral. Then briefly explain why this integral is difficult to compute. (b) (8 points) Describe the solid E using cylindrical coordinates.Then express the mass of E as...
(For 5b, please use the y-axis as the axis of symmetry for the
cylinder)
5) a-b Set-up the flux integrals for the given surfaces in the variables indicated. Your final answer should be a scalar- valued double integral. That is, the double integral should does not contain any vector quantities. The differential is given. Do not solve the integrals you setup in a. and b. No work is needed for a-b. a. F(x, y, z) = 5î + 10ủ +...
Use rectangular, cylindrical and spherical coordinates to set up the triple integrals representing the volume of the region bounded below by the xy plane, bounded above by the sphere with radius and centered at the origin the equation of the sphere is x2 + y2 + z2-R2), and outside the cylinder with the equation (x - 1)2 +y2-1 (5 pts each) Find the volume by solving one of the triple integrals from above.( 5 pts) Total of 20 pts)
Use...
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) (6 points) Find an expression for the mass of E as a triple integral. Then briefly explain why this integral is difficult to compute Suppose E is the half-cylinder described by r' +ґ-1 between z = 4 and the xy-plane where y > 0....
11. Evaluate S. 'S*(1 + 3x2 + 2y?) dx dy. 12. Find the volume in the first octant of the solid bounded by the cylinder y2 + z2 = 4 and the plane x = 2y. Graph for Problem 12 13. Find the volume under the paraboloid z = 4 - x2 - y2 and above the xy-plane. N Consider the solid region bounded above by the sphere x + y + z = 8 and bounded below by the...
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,...
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...