Find the average value of the function f over the given region.f(x, y) = ; R = {(x, y): 1 ≤ x ≤ 3, 1 ≤ y ≤ 3}

Solve the problem.Integrate over the region 0 ≤ x2 + y2 ≤ 36.

Solve the problem.Let D be the smaller cap cut from a solid ball of radius 9 units by a plane 4 units from the center of the sphere. Set up the triple integral for the volume of D in cylindrical coordinates.

Sketch g(D) from the description of D and change of variables  x = u + v, y = 2u – v where D is the rectangle 0 ≤ u ≤ 1, -1 ≤ v ≤ 1

Solve the problem.Let D be the smaller cap cut from a solid ball of radius 6 units by a plane 5 units from the center of the sphere. Set up the triple integral for the volume of D in cylindrical coordinates.

Solve the problem.Integrate over the region 0 ≤ x2 + y2 ≤ 81.

Solve the problem.Integrate over the region 1 ≤ x2 + y2 ≤ 16.

Find the volume of the indicated region.the tetrahedron cut off from the first octant by the plane

Find the Jacobian J(u, v).

Sketch g(D) from the description of D and change of variables x = 3u, y = v where D is the rectangle 0 ≤ u ≤ 2, 1 ≤ v ≤ 4