Question

Use spherical coordinates.

(a) Find the centroid of a solid homogeneous hemisphere of radius 1. (Assume the upper hemisphere of a sphere centered at the origin. Use the density function

*ρ*(* x*,

(*x*, *y*, *z*) =

(b) Find the moment of inertia of the solid in part (a) about a
diameter of its base.

*I _{d}* =

Answer #1

Use spherical coordinates. Evaluate (6 − x^2 − y^2) dV, where H
is the solid hemisphere x^2 + y^2 + z^2 ≤ 16, z ≥ 0.

A solid sphere, radius R, is centered at the origin. The
“northern” hemisphere carries a uniform charge density ρ0, and the
“southern” hemisphere a uniform charge density −ρ0. Find the
approximate field E(r,θ) for points far from the sphere (r ≫
R).

3. Consider a solid hemisphere of radius R, constant mass
density ρ, and a total mass M. Calculate all elements of the
inertia tensor (in terms of M and R) of the hemisphere for a
reference frame with its origin at the center of the circular base
of the hemisphere. Make sure to clearly sketch the hemisphere and
axes positions.

Use spherical coordinates. Evaluate (2 − x2 − y2) dV, where H is
the solid hemisphere x2 + y2 + z2 ≤ 25, z ≥ 0. H

Use spherical coordinates.
I=exp[-(x2+y2+z2)3/2;
E=upper hemisphere of radius 2

Use the triple integrals and spherical coordinates to find the
volume of the solid that is bounded by the graphs of the given
equations. x^2+y^2=4, y=x, y=sqrt(3)x, z=0, in first octant.

) Use spherical coordinates to find the volume of the solid
situated below x^2 + y ^2 + z ^2 = 1 and above z = sqrt (x ^2 + y
^2) and lying in the first octant.

A solid is described along with its density function. Find the
center of mass of the solid using cylindrical coordinates:
The upper half of the unit ball, bounded between z = 0 and z =
√(1 − x^2 − y^2) , with density function δ(x, y,z) = 1.

In spherical coordinates, find the volume of the region bounded
by the
sphere x^2 + y^2 + z^2 = 9 and the plane z = 2.

Find the rectangular coordinates of the point whose spherical
coordinates are given. (a) (1, 0, 0) (x, y, z) = (b) (14, π/3, π/4)
(x, y, z) =

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