Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have several tasks to solve where a set of inequalities is used to describe a region. I should then calculate the area or volume of that region.

Let's say we have the following inequality (for $x,y,z \geq 0$):

$x+2y+3z \leq 1$

Now I need to find out the boundaries for the triple integral. From a few examples I have here, I wasn't able to derive a way to solve such a problem.

The example solution suggests that

$0 \leq y \leq \frac{1-x}{2}$

and

$0 \leq z \leq \frac{(x - 2y)}{3}$.

for

$0 \leq x \leq 1$

leading to the integral:

$ \int_{0}^{1} \int_{0}^{\frac{1-x}{2}} \int_{0}^{\frac{x-2y}{3}} { 1 \; dz dy dx } $

How do I find out these boundaries? (Moreover, how do I find out the boundaries in a general approach)

PS: I'm not sure if the solution is correct at all, that's probably why I'm confused.

share|improve this question
    
Is the upper bound for $z$ really correct? The integral evaluates to 0. –  Calle Jan 19 '11 at 13:26
    
The upper bound for $z$ should be $(1-x-2y)/3$. –  Hans Lundmark Jan 19 '11 at 13:45
add comment

2 Answers

up vote 5 down vote accepted

Take the variables one by one.

For your example, start with $z$ and move all the other variables to the other side:

$0 \leq 3z \leq 1 - (x+2y)$

Divide by $3$ and you have your bound.

For $y$, disregard $z$, but do the same thing, obtaining:

$0 \leq 2y \leq 1 - x$

Same thing for $x$ immediately gives

$0 \leq x \leq 1$.

I think the general approach is pretty evident from this.

With these limits, you get the integral

$\int_0^1 \int_0^\frac{1-x}{2} \int_0^\frac{1-x-2y}{3} dzdydx = \frac{1}{36}$

EDIT: This should be correct, since the volume described can be seen as a pyramid with base area $\frac{1}{12}$ (triangle with height $\frac{1}{2}$, base $\frac{1}{3}$) and height 1. The volume of a pyramid is $\frac{1}{3}Bh$.

share|improve this answer
    
Thank you very much, that made it very clear (at least for one inequality) and probably saved me from failing a test today. However, in the case of multiple inequalities, should I combine them first? –  slhck Jan 19 '11 at 13:41
1  
Depends on what the inequalities look like. Draw a picture. It's best if you give an example. –  Calle Jan 19 '11 at 14:07
    
Okay, I think I got the hang of it. Drawing the picture really helps with seeing the boundaries. –  slhck Jan 19 '11 at 15:59
add comment

always draw a picture! you'll see the volume lying between the $xy$-plane and the plane $x+2y+3z=1$, a tetrahedron. It's projection on the $xy$-plane is a right triangle with hypotenuse $x+2y=1$ (obtained by setting $z=0$). IMHO, the geometric approach is better than messing with a bunch of inequalities!

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.