Question

B is the solid occupying the region of the space in the first octant and bounded by the paraboloid z = x2 + y2- 1 and the planes z = 0, z = 1, x = 0 and y = 0. The density of B is proportional to the distance at the plane of (x, y).

Determine the coordinates of the mass centre of solid B.

Answer #1

a) Sketch the solid in the first octant bounded by:
z = x^2 + y^2 and x^2 + y^2 = 1,
b) Given
the volume density which is proportional to the distance from the
xz-plane, set up integrals
for finding the
mass of the solid using cylindrical
coordinates, and spherical coordinates.
c) Evaluate one of these to find the mass.

a. Let S be the solid region first octant bounded by the
coordinate planes and the planes x=3, y=3, and z=4 (including
points on the surface of the region). Sketch, or describe the shape
of the solid region E consisting of all points that are at most 1
unit of distance from some point in S. Also, find the volume of
E.
b. Write an equation that describes the set of all points that
are equidistant from the origin and...

Prove Gauss's Theorem for vector field F= xi +2j +
z2k, in the region bounded by planes z=0, z=4, x=0, y=0
and x2+y2=4 in the first octant

Let D be the solid in the first octant bounded by the planes
z=0,y=0, and y=x and the cylinder 4x2+z2=4.
Write the triple integral in all 6 ways.

4. Consider the solid bounded by the paraboloid x^2+ y^2 + z = 9
as well as by the planes y = 3x and z = 0 in the first octant.
(a) Graph the integration domain D.
(b) Calculate the volume of the solid with a double
integral.

Question 2
D is the region in the first octant bounded by: z = 1 −
x2 and z = ( y − 1 )2
Sketch the domain D.
Then, integrate f (x, y, z) over the domain in 6 ways: orderings of
dx, dy, dz.

Determine the centroid, C(x̅, y̅, z̅), of the solid formed in
the first octant bounded by y = 4 − x^2 and x − z = 0.

Determine the centroid C(x,y,z) of the solid formed in
the first octant bounded by z+y-16=0 and x^2=16-y.

Use cylindrical coordinates to find the volume of the region
in the first octant bounded by a cylinder ?^2+ ?^2= 9 and a plane
2? + 3? + 4? = 12.

Find the center of mass of the region bounded by the paraboloid
x^2 + y^2 − 2 = z and the plane x + y + z = 1 assuming the region
has uniform density 8.

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