Question

- Determine the volume of the tetrahedron cut from the first octant by the plane 6x + 2y + z = 6. Sketch the solid.

Answer #1

Find 6 different iterated triple integrals for the volume of the
tetrahedron cut from the first octant (when x > 0, y > 0, and
z > 0) by the plane 6x + 2y + 3z = 6. Dont evaluate the
integrals.

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

1) Use calculus to find the volume of the solid pyramid in the
first octant that is below the planes x/ 3 + z/ 2 = 1 and y /5 + z
/2 = 1. Include a sketch of the pyramid.
2)Find three positive numbers whose sum is 12, and whose sum of
squares is as small as possible, (a) using Lagrange multipliers
(b )using critical numbers and the second derivative test.

Use a triple integral to find the volume of the given solid.
The tetrahedron enclosed by the coordinate planes and the
plane
11x + y + z = 2

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.

Find the surface area of the portion of the plane 3x+2y+z=6 that
lies in the first octant

1- Set up the triple integral for the volume of the sphere Q=8
in rectangular coordinates.
2- Find the volume of the indicated region.
the solid cut from the first octant by the surface z= 64 - x^2
-y
3- Write an iterated triple integral in the order dz dy dx for
the volume of the region in the first octant enclosed by the
cylinder x^2+y^2=16 and the plane z=10

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 y = 4 − x^2 and x − z = 0.

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...

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