Question

Calculate the volume bounded by the plane x + 2y + 3z = 6 by coordinate planes with a triple integral.

Answer #1

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

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.

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
using double integrals.

The tetrahedron is the first octant bounded by the coordinate
planes and the plane passing through (1,0,0), (0,2,0), and
(0,0,3).
I need to calculate the volume of this region, how should this
be done?

Use Lagrange multipliers to find the point on the
plane
x − 2y + 3z = 6
that is closest to the point
(0, 2, 5).
(x, y, z) =

Find the volume of the solid bounded by the surface z=
5+(x-y)^2+2y and the planes x = 3, y = 3 and coordinate planes.
a. First, find the volume by actual calculation.
b. Estimate the volume by dividing the region into nine equal
squares and evaluating the functional value at the mid-point of the
respective squares and multiplying with the area and summing it.
Find the error from step a.
c. Then estimate the volume by dividing each sub-square above...

Consider the plane with general equation x - 2y - 3z = 6. Which
one of the following equation for a line that does not intersect
this plane?
a. (x, y, z) = (1, -2, -3) + t(1, -1, 1), t ∈ R
b. (x, y, z) = (1, -2, -3) + t(1, -2, -3), t ∈ R
c. (x ,y, z) = (1, 2, -3) + t(1, -1, 1), t ∈ R
d. (x, y, z) = (1, 2,...

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

Given the parallel planes x + 2y + 3z = 1 and 3x + 6y + 9z = 18.
Find a normal vector to these planes.

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.

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