Evaluate the triple integral. 2xy dV, where E is bounded by the parabolic cylinders y = x² and x = y2 and the planes z = 0 and z = 5x + y

A solid lies inside the sphere x^2 + y^2 + z^2 = 6z and outside the cone z=sqrt(x^2+y^2). Write a description of the solid in terms of inequalities involving spherical coordinates

et △ABC be a triangle in S^2 (the two-dimensional sphere). Let the dual point C'∈S^2 of C be defined by the following three conditions:(i) d(C' , A)=π/2(ii) d(C' , B)=π/2(iii) d(C' , C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, andit follows (you may assume) that △A' B' C' is too.(question) Are there triangles △ABC in S^2 identical to their own dual △A'B'C'?   I would be very thankful if you could provide some explanation with the steps, thank you in advance.

Which surface below is an elliptic paraboloid? O x^2 + y^2/4 - z^2 = 1 O z = -sqrt(x^2 + y^2) z = -x^2/4 - y^2 O z = y^2 - x^2/9

Solve for the distance between the point P(0,0,0,0) and the plane w + x + y + z = 3 in R^4.

How do I find the surface area of the sphere z=(9-(x^2)-(y^2))^(1/2) inside the cylinder (x^2)+(y^2)=1?

Use the polar coordinates to find the surface area of the part of the conical surface z^2 = x^2 + y^2, that lies above the semi-circle y = sqrt(2-x^(2))   (the semicircle lies on teh xy-plane)

the voulme of the region are  bounded by the paraboloid x= y^2+z^2 and the half cone x= 8sqrt(y^2+z^2) is

Instead of doing disk washer method to find volume for this. How can you find volume using vector calculus on this particular problem? Th

What is the shortest distance from the surface xy+3x+z^2=12 to the origin?