Let ∭E (yz)dV, where E = {(x,y,z)/ x = 1 - y^2 - z^2, x>=0} a. Sketch E, the solid of integrat...
Let ∭E (yz)dV, where E = {(x,y,z)/ x = 1 - y^2 - z^2, x>=0}
a. Sketch E, the solid of integration.
b. Sketch D, the region of integration in the plane the solid is projected onto.
c. Evaluate the integral using cylindrical coordinates.
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Let ∭E (yz)dV, where E = {(x,y,z)/ x = 1 - y^2 - z^2, x>=0} a. Sketch E, the solid of integrat...
10. Consider the integral (x + y + z) dV where D is the volume inside...
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...
Q3. Sketch the region of integration for the integral [5(8,19,2) dr dz dy. (2, y, z)...
Q3. Sketch the region of integration for the integral [5(8,19,2) dr dz dy. (2, y, z) do dzdy. Write the five other iterated integrals that are equal to the given iterated integral. Q4. Use cylindrical coordinates and integration (where appropriate) to complete the following prob- lems. You must show the work needed to set up the integral: sketch the regions, give projections, etc. Simply writing out the iterated integrals will result in no credit. frs:52 (a) Sketch the solid given...
6. (12pts) Consider the solid that is above the xy-plane, bounded above by =/4-x-y and below by +y a. Sketch the so...
6. (12pts) Consider the solid that is above the xy-plane, bounded above by =/4-x-y and below by +y a. Sketch the solid formed by the given surfaces b. Set up in rectangular coordinates the triple integral that represents the yolume of the solid. Sketch the appropriate projection. Do NOT evaluate the integrals. (Hint: Let dV- d dy de) c. Set up in cylindrical coordinates the triple integral that represents the volume of the solid. Sketch the appropriate projection. Do NOT...
5a. The solid E lies above the cone z =V3V2 + y and below the sphere...
5a. The solid E lies above the cone z =V3V2 + y and below the sphere cº + y2 + 2 = 9. Completely set up, but DO NOT EVALUATE, the triple integral Ssse (y+z)dV in spherical coordinates. Show appropriate work for obtaining the limits of integration and include a sketch. Your answer should be completely ready to evaluate. (9 points) 5b. Completely set up, but DO NOT EVALUATE, the same triple integral ple (y + x)2V from part (a),...
1. Let V be the solid region in R3 that lies within the sphere 2+y+z2-4, above the zy-plane, and ...
please help with Q1 and 3
1. Let V be the solid region in R3 that lies within the sphere 2+y+z2-4, above the zy-plane, and below the cone z -Vx2 + y2 (a) Sketch the region V (b) Calculate the volume of V by using spherical coordinates. (c) Find the surface area of the part of V that lies on the sphere z2 y 24, by calculatinga surface integral. (d) Verify your solution to (c) by calculating the surface integral...
Please explain steps 3. Consider the triple integral , g(x, y, z)dV, where E is the...
Please explain steps
3. Consider the triple integral , g(x, y, z)dV, 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 (0,0,0).
2. (13 points) Let E be the solid region bounded by the planes x = 0,...
2. (13 points) Let E be the solid region bounded by the planes x = 0, y = 0, 2=0, and x+y+z=1. (a) Sketch E. (b) Set up the integral SSSe ex+y+z dV as a triple iterated integral. (c) Compute the integral.
For the described solid S, write the triple integral f(x,y, z)dV as an iterated integral in (i) r...
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...
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-axi...
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...
[4] Let Q be the solid region in space below the plane z = 4, outside...
[4] Let Q be the solid region in space below the plane z = 4, outside the cylinder x2 + y2 = 1, and above the paraboloid z = x2 + y2 (see figure). 1 Express the integral =dV as an iterated integral in Ida x² + y² +2² cylindrical coordinates. Do not evaluate the integral.
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...
Q3. Sketch the region of integration for the integral [5(8,19,2) dr dz dy. (2, y, z) do dzdy. Write the five other iterated integrals that are equal to the given iterated integral. Q4. Use cylindrical coordinates and integration (where appropriate) to complete the following prob- lems. You must show the work needed to set up the integral: sketch the regions, give projections, etc. Simply writing out the iterated integrals will result in no credit. frs:52 (a) Sketch the solid given...
6. (12pts) Consider the solid that is above the xy-plane, bounded above by =/4-x-y and below by +y a. Sketch the solid formed by the given surfaces b. Set up in rectangular coordinates the triple integral that represents the yolume of the solid. Sketch the appropriate projection. Do NOT evaluate the integrals. (Hint: Let dV- d dy de) c. Set up in cylindrical coordinates the triple integral that represents the volume of the solid. Sketch the appropriate projection. Do NOT...
5a. The solid E lies above the cone z =V3V2 + y and below the sphere cº + y2 + 2 = 9. Completely set up, but DO NOT EVALUATE, the triple integral Ssse (y+z)dV in spherical coordinates. Show appropriate work for obtaining the limits of integration and include a sketch. Your answer should be completely ready to evaluate. (9 points) 5b. Completely set up, but DO NOT EVALUATE, the same triple integral ple (y + x)2V from part (a),...
please help with Q1 and 3
1. Let V be the solid region in R3 that lies within the sphere 2+y+z2-4, above the zy-plane, and below the cone z -Vx2 + y2 (a) Sketch the region V (b) Calculate the volume of V by using spherical coordinates. (c) Find the surface area of the part of V that lies on the sphere z2 y 24, by calculatinga surface integral. (d) Verify your solution to (c) by calculating the surface integral...
Please explain steps
3. Consider the triple integral , g(x, y, z)dV, 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 (0,0,0).
2. (13 points) Let E be the solid region bounded by the planes x = 0, y = 0, 2=0, and x+y+z=1. (a) Sketch E. (b) Set up the integral SSSe ex+y+z dV as a triple iterated integral. (c) Compute the integral.
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...
[4] Let Q be the solid region in space below the plane z = 4, outside the cylinder x2 + y2 = 1, and above the paraboloid z = x2 + y2 (see figure). 1 Express the integral =dV as an iterated integral in Ida x² + y² +2² cylindrical coordinates. Do not evaluate the integral.
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