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
Attachments
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.