Question

- 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.

Answer #1

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.

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.

Let E be the solid that lies in the first octant, inside the
sphere x2 + y2 + z2 = 10. Express the volume of E as a triple
integral in cylindrical coordinates (r, θ, z), and also as a triple
integral in spherical coordinates (ρ, θ, φ). You do not need to
evaluate either integral; just set them up.

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.

Lets consider the solid bounded above a sphere x^2+y^2+z^2=2 and
below by the paraboloid z=x^2+y^2.
Express the volume of the solid as a triple integral in
cylindrical coordinates. (Please show all work clearly) Then
evaluate the triple integral.

The region is bounded by y=2−x^2 and y=x. (a) Sketch the region.
(b) Find the area of the region. (c) Use the method of cylindrical
shells to set up, but do not evaluate, an integral for the volume
of the solid obtained by rotating the region about the line x = −3.
(d) Use the disk or washer method to set up, but do not evaluate,
an integral for the volume of the solid obtained by rotating the
region about...

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...

4. Let W be the three dimensional solid inside the sphere x^2 +
y^2 + z^2 = 1 and bounded by the planes x = y, z = 0 and x = 0 in
the first octant. Express ∫∫∫ W z dV in spherical coordinates.

A solid E ib tge furst ictabt us viybded aboce by the sohere
x^2+y^2+z^2=4 , lateral by the cylinder x^2+y^2=1, and by the
coordinate olanes. Set up the integral SSS (x^2+y^2+z^2) dV in a)
rectangular, b) cylindrical, and c) spherical coordinates. You do
not need to evaluate any of the integrals. Show clearly how you
come up with the limits of integrations where necessary.

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.

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