Question

Let D be the solid in the first octant bounded by the planes
z=0,y=0, and y=x and the cylinder 4x^{2}+z^{2}=4.
Write the triple integral in all 6 ways.

Answer #1

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...

6. Let R be the tetrahedron in the first octant bounded by the
coordinate planes and the plane passing through (1, 0, 0), (0, 1,
0), and (0, 0, 2) with equation 2x + 2y + z = 2, as shown below.
Using rectangular coordinates, set up the triple integral to find
the volume of R in each of the two following variable orders, but
DO NOT EVALUATE.
(a) triple integral 1 dxdydz
(b) triple integral of 1 dzdydx

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.

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.

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.

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.

Find the integral that represents the volume of the solid
bounded by the planes y = 0, z = 0, y = x, and 6x + 2y + 3z = 6. No
need to solve the integral.

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.

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.

Find the center mass of the solid bounded by planes x+y+z=1, x=0
y=0, and z=0, assuming a mass density of ρ(x,y,z)=7sqrt(z)
Xcm
Ycm
Z cm

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