(9 points) Suppose f(x, y, z) = - and D is the domain inside the sphere...
The region is a cone, z == ? + ytopped by a sphere of radius 4. Find the limits of integration on the triple integral for the volume of the snowcone using Cartesian, cylindrical, and spherical coordinates and the function to be integrated. For your answers 0 = theta, o =phi, and p = rho. Cartesian V= "SC"}, "plz,y,z2) dz dydz where A B = .D= and p(x, y, z) = E= F= Cylindrical v=L" S "*P10,0,2)dz dr do where...
10. Consider the integral (x + y + z) dV where D is the volume inside the sphere x2 + y2 + x2 = 9 and above the plane z = 1. (a) (3 marks) Express I as an iterated integral using Cartesian coordinates with the order of integration z, x and y. DO NOT EVALUATE THIS INTEGRAL. (b) (3 marks) Express I as an iterated integral using spherical coordinates with the order of integration p, 0, and 0. DO...
Question 8.6. The solid inside the sphere x? + y2 + 2? 3 4 and outside the cylinder I TY has density f(x, y, z) = typ • Write a triple integral (including the limits of integration) in cylindrical coordinates that gives the mass of this solid. • Write a triple integral (including the limits of integration) in spherical coordinates that gives the mass of this solid • Compute the mass of the solid using the integral that seems easier...
(a) Set up a double integral for calculating the flux of the vector field F(x, y, z) = z2k through the upper hemisphere of the sphere x2 + y2 + z2 = 4, oriented away from the origin. If necessary, enter P as rho, 8 as theta, and o as phi. B D Flux IT do de А A= B= C = D= (b) Evaluate the integral. Flux = F.dĀ= SI S
For the described solid S, write the triple integral f(x,y, z)dV as an iterated integral in (i) rectangular coordinates (x,y, z); (ii) cylindrical coordinates (r, 0, 2); (iii) spherical coordinates (p, φ,0). a. Inside the sphere 2 +3+224 and above the conezV b. Inside the sphere x2 + y2 + 22-12 and above the paraboloid z 2 2 + y2. c. Inside the sphere 2,2 + y2 + z2-2 and above the surface z-(z2 + y2)1/4 d. Inside the sphere...
5. Express the triple integral | f(x,y,z)dV as an iterated integral in cartesian coordinates. E is the region inside the sphere x2 + y2 + z2 = 2 and above the elliptic paraboloid z = x2 + y2
(16) (7 points) Set up an iterated integral of f (x, y, z) = x2 + y2 + z2 over the solid region shown below. Use the spherical coordinates. N 1 y - One-eighth sphere
The region is a right circular cone, 2 = Var? + y2 with height 29. Find the limits of integration on the triple integral for the volume of the cone using Cartesian, cylindrical, and spherical coordinates and the function to be integrated. For your answers 0 theta, o=phi, and p = rho. Cartesian V = p(x, y, z) dz dy da where A C = B = ,F= E ,D= and p(x, y, z) = Cylindrical V = so" C"S"...
Question 7 5 pts each Write iterated integrals for each of the given calu lations. Do not evaluate. (A) The integral of f(x, )212y over the domain D: 2 y 20. (B) The integral of f(x, y, z) = 12x + 3 over the volume contained in the first octant and below the graph z 8-y 2 (C) The mass of an object occupying the region bounded between the sur faces x2 + y2 + Z2 = 16 and z...
Consider the triple integral SISE g(x,y,z)d), where E is the solid bounded above by the sphere x2 + y2 + z2 = 18 and below by the cone z? = x2 + y2. a) Set up the triple integral in rectangular coordinates (x,y,z). b) Set up the triple integral in cylindrical coordinates (r, 0,z). c) Set up the triple integral in spherical coordinates (2,0,0).