Solve the initial value problem.Differential Equation:  = (3t2 + 2t)i + (4t3+ 9)jInitial Condition: r(0) = 6j

Find the volume of the indicated region.the region that lies under the plane z = 3x + 6y and over the triangle with vertices at (1, 1), (2, 1), and (1, 2). Give an EXACT answer, no decimal approximations. If necessary, use the forward slash, /, for fractions.

Use the given transformation to evaluate the integral.

Solve the problem.Let R ={(x, y): 1 ≤ x < 4, 0 ≤ y < 4}. Evaluate  

Find the average value of F(x, y, z) over the given region.F(x, y, z) = x9y10z2 over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 2, y = 1, z = 3.

Use cylindrical coordinates to find the volume of the indicated region.the region enclosed by the paraboloids z = x2 + y2 – 9 and z = 153 – x2 – y2.

Find the transformation from the uv-plane to the xy-plane.,    

Find the volume by using polar coordinates.The region bounded by the paraboloid z = x2 + y2, the cylinder x2 + y2 = 49, and the xy-plane.

Use cylindrical coordinates to find the volume of the indicated region.the region enclosed by the paraboloids z = x2 + y2 – 9 and z = 63 – x2 – y2.  

Find the volume by using polar coordinates.The region bounded by the paraboloid z = x2 + y2 ,  the cylinder x2 + y2 = 64, and the xy-plane.