Volumes with triple integrals, aka I suck at geometry

In summary, the conversation discusses finding the volume of a bounded body by using variable substitutions and setting up integrals. The problem involves four planes and the individual steps are outlined. The conversation also touches on the possibility of needing calculus and finding the vertices of the shape.
  • #1
Gauss M.D.
153
1

Homework Statement



Calculate the volume of the body that is bounded by the planes:

x+y-z = 0
y-z = 0
y+z = 0
x+y+z = 2

Homework Equations





The Attempt at a Solution



I made a variable substitution

u = y+z
v = y-z
w = x

which gave me the new boundaries

u+w = 2
v+w = 0
v = 0
u = 0

Problem is, I must have slept late the day they taught this is class. What do I do to determine which way these inequalities go?!?
 
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  • #2
Gauss M.D. said:

Homework Statement



Calculate the volume of the body that is bounded by the planes:

x+y-z = 0
y-z = 0
y+z = 0
x+y+z = 2

Do you think you need calculus?

$$z=x+y \\ z= -x -y +2 \\ z=y \\ z=-y$$

That should be pretty simple...:tongue:
 
  • #3
You should not change variables before you have set up the integral. Do that first.
 
  • #4
To get an idea of the shape, try to find the vertices.
 

1. What is a triple integral?

A triple integral is a mathematical concept that is used to calculate the volume of a three-dimensional shape. It involves integrating a function over three variables, usually represented by x, y, and z.

2. How is a triple integral different from a regular integral?

A regular integral is used to calculate the area under a curve in two dimensions, while a triple integral is used to calculate the volume under a surface in three dimensions. Triple integrals involve integrating over three variables, while regular integrals only involve one variable.

3. Why is it called a "triple" integral?

The term "triple" comes from the fact that a triple integral involves integrating over three variables. It is also sometimes referred to as a "volume integral" because it is used to calculate the volume of a three-dimensional shape.

4. What types of shapes can be calculated using triple integrals?

Triple integrals can be used to calculate the volume of any three-dimensional shape, such as cubes, spheres, cones, cylinders, and more complex shapes.

5. How can I improve my understanding of triple integrals?

Practice is key when it comes to understanding triple integrals. You can also seek out resources such as textbooks, online tutorials, and practice problems to help you better understand the concept and improve your skills.

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