# Vectors Calculas & Applications

See the attached file.

1. Evaluate the integral:

** see the attachment for the full equation **

Where D is the domain given by: 1 ≤ x^2 + y^2 ≤ 4 and y ≥ 0.

2. (i) Find the area of the region enclosed by the ellipse (x^2/a^2) + (y^2/b^2) = 1

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

Sketch the region.

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

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

4. Evaluate the moment of inertia of a cylinder of radius R and height L about its axis of symmetry, if the density varies with distance from the axis as p = a(x^2 + y^2). Express the result in terms of the cylinder mass.

5. Calculate the volume of the ellipsoid

(x^2/a^2) + (y^2/b^2) + (z^2/c^2) ≤ 1.

https://brainmass.com/math/vector-calculus/vectors-calculas-applications-529056

#### Solution Preview

Please see the attached file for the complete solution.

1. We wish to evaluate the integral:

(please see the attached file)

where D is the domain given by (please see the attached file) and (please see the attached file)

The easiest way to solve this integral is to transform to polar coordinates. We have:

(please see the attached file)

2. (i) We wish to find the area A of the region D enclosed by the ellipse:

(please see the attached file)

For every point in D, we have:

(please see the attached file) ...

#### Solution Summary

In this solution we solve several problems involving multiple integrals.