6. Let R be the tetrahedron in the first octant bounded by the
coordinate planes 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
B is the solid occupying the region of the space in the first
octant and bounded...
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.
A solid Tetrahedron has vertices (0, 0, 0), (2, 0, 0), (0, 4,
0), and (0,...
A solid Tetrahedron has vertices (0, 0, 0), (2, 0, 0), (0, 4,
0), and (0, 0, 6).
(a) i. Sketch the tetrahedron in the xyz-space.
ii. Sketch (and shade) the region of integration in the
xy-plane.
(b) Setup one double integral that expresses the volume of the
tetrahedron. Define the proper limits of integration and the order
of integration. DO NOT EVALUATE.
Determine the centroid, C(x̅, y̅, z̅), of the solid formed in
the first octant bounded by...
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.
a. Let S be the solid region first octant bounded by the
coordinate planes and the...
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...