How to determine the volume of a region bounded by planes?

• Tom31415926535
Remember to integrate over the correct order. In summary, the conversation discusses finding the volume and center of mass of a region bounded by several planes using triple integration. The first step is to draw a sketch and determine the boundaries for each variable. The boundaries for this particular region are 0≤z≤x+y, 0≤y≤1-x, and 0≤x≤1.
Tom31415926535

Homework Statement

Let G be the region bounded by the planes x=0,y=0,z=0,x+y=1and z=x+y.

Homework Equations

(a) Find the volume of G by integration.
(b) If the region is a solid of uniform density, use triple integration to find its center of mass.

The Attempt at a Solution

[/B]
My understanding is that I need to setup a triple integral:

∫∫∫dxdydz

I’m just a little unsure about how to determine the terminals

Did you draw a sketch?

You can always find boundaries for e.g. z as function of x,y and boundaries for y as function x, but sometimes there are easier methods.

Have you drawn a picture? That's the first step. Maybe this will help:

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mfb said:
Did you draw a sketch?

You can always find boundaries for e.g. z as function of x,y and boundaries for y as function x, but sometimes there are easier methods.
So would the boundaries be:

0≤z≤x+y
0≤y≤1-x
0≤x≤1

That will work.

1. What is the formula for finding the volume of a region bounded by planes?

The formula for finding the volume of a region bounded by planes is V = ∫∫∫dV, which represents the triple integral of the region.

2. How do you determine the region bounded by planes?

To determine the region bounded by planes, you first need to find the intersection points of the planes. Then, use these points to create a boundary for the region. The region will be bounded by these planes and can be visualized as a solid shape.

3. Can the region bounded by planes have a curved shape?

No, the region bounded by planes will always have a flat, polygonal shape. This is because the planes themselves are flat surfaces and the region is defined by the intersection of these planes.

4. What are the units of the volume of a region bounded by planes?

The units of the volume of a region bounded by planes will depend on the units used for the boundaries of the region. For example, if the boundaries are in meters, the volume will be in cubic meters.

5. Is it possible to determine the volume of a region bounded by planes if the boundaries are not perpendicular to each other?

Yes, it is still possible to determine the volume of a region bounded by planes even if the boundaries are not perpendicular to each other. This can be done by using the appropriate formula for the triple integral, which takes into account the angle between the planes.

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