what is the volume?

Question

hey guys....

i have a hard time answering this problem.....

here it goes...

use the double integral to find the volume of a solid bounded by the surface x + 2y + 3z = 6 and the coordinate planes....

GOOD LUCK guys....

it's in our finals exam and i have no idea bout this one.....

can you help me?

have a nice day guys!!!!

jason

Answer 1

First, note that this plane intersects all three axes at positive coordinates, so the solid bounded by this plane and the coordinate planes will be entirely within the first octant [+, +, +]. This means that x≥0, y≥0, and z≥0. For the latter two conditions to hold, this means x≤6. So the range for the x-coordinate is [0, 6]. Now, for this to hold for any particular x-coordinate, y cannot exceed (6-x)/2, so the range for y is [0, 3 - x/2]. Finally, at any given (x, y) coordinate, z ranges from 0 to (6-x-2y)/3. So the height of the solid at that point is thus (6-x-2y)/3. Placing this into an integral, we have:

[0, 6]∫ [0, 3-x/2]∫ (6-x-2y)/3 dy dx.

Now we just integrate:

[0, 6]∫ (6y-xy-y²)/3 | [0, 3-x/2] dx
[0, 6]∫ (6(3-x/2) - x(3-x/2) - (3-x/2)²)/3 dx

Simplifying:

[0, 6]∫ (18 - 3x - 3x + x²/2 - (x²/4 - 3x + 9))/3 dx
[0, 6]∫ (9 - 3x + x²/4)/3 dx
[0, 6]∫x²/12 - x + 3 dx

x³/36 - x²/2 + 3x | [0, 6]
6 - 18 + 18
6

Answer 2

It looks like a tetrahedron with vertices at:

(0, 0, 0)
(6, 0, 0)
(0, 3, 0)
(0, 0, 2)

The following shows how to compute the volume of a tetrahedron:

http://mathcentral.uregina.ca/QQ/database/QQ.09.03/peter2.html

yielding (1/3)Bh with B on the z=0 plane being (1/2) (6) (3) and h = 2 so volume is 18 [D'oh! Correction after seeing Pascal's reply: (1/3) * (1/2) (6) (3) (2) = 6, not 18]. Use that to check your double integral, which could be any of the six (two of the three variables, in either order), for example

outer: x from 0 to 6
inner: y from 0 to (6 - x)/2
of: dy dx (height from z=0 to surface x + 2y + 3z = 6, which is (6 - 2y - x) / 3).

or

outer z: from 0 to 2
inner y: from 0 to (6 - 3z)/2
of: (6 - 2y - 3z) dy dz

etc.

Answer 3

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