Vectors calculas and applications

Question

Question:

Vectors Calculas & Applications

1. Evaluate the integral:

                    I = ∫∫_D dxdy/(x²+y²)^5/2.

   where D is the domain given by: 1<x2 + y2 < 4 and y>0.

2. (i) Find the area of the region enclosed by the ellipse x²/a² + y²/b² = 1

    (ii) Find the area of the region enclosed by the parabola y = x2 and the line y=x+2. Sketch the region.

                                               Triple integrals

3. Calculate the mass of a spherical bead of radius 3, centered at the origin, if the destiny of the material it is made of, is given by the function:

                                      ρ(x,y,z) = 2|z|

4. . Evaluate the moment of inertia of a cylinder of radius R and hight L about its axis of symmetry, if the density varies with distance from the axis as ρ = a(x2 + y2). Express the result in terms of the cylinder mass.

5. Calculate the volume of the ellipsoid:          x²/a² + y²/b² + z²/c² < 1.