f (x, y, z)= xy; s is the plane x=2-x-y in the first octant f(x, y, z)=^2+y^2; s

Question

f (x, y, z)= xy; s is the plane x=2-x-y in the first octant f(x, y, z)=^2+y^2; s is the paraboloid z=x^2+y^2, for 0 z 4 f(x, y, z)= 25-x^2-y^2; S is the hemisphere centered at the origin with radius 5, for z 0 Find the average temperature on that part of the plane t + 4y + z - 6 over the square |x| s I. |y| s I. where the temperature is given by T(x ,y ,z) =e^-z Find the average squared distance between the origin and the points on the paraboloid z = 4 - x2 - y2, for z ^ 0. Find the average value of the function f{x, y, z) = jry: on the unit sphere in the first octant. Find the average value of the temperature function ( T(x,y, z) = 100 - 25z on the cone z^2 = x^2 + y^2, for Surface integrals of vector fields Find the flux of the follow ing vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface. f = (0, 0, - 1) across the slanted face of the tetrahedron z = 4 - x - y in the first octant; normal vectors point upward.

Explanation / Answer

since it is given unit sphere so, x^2 + y^2 + z^2 <= 1

now by using sherical coordinates we will get ,

xyz dV =
R

2 . . . . . . . . . . . . . . . . . 1
cossin d sin^3 d ^5 d = 0

0 . . . . . . . . . . 0 . . . . . . . . 0

so we will have it   zero     answer

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