The Operation of Multiple Ranges in Definite Integration

[Hypatio]

I have the equation
$$G[xy\textup{ln}(z+R)+xz\textup{ln}(y+R)+yz\textup{ln}(x+R)]|^{x_2-x_0}_{x_1-x_0}|^{y_2-y_0}_{y_1-y_0}|^{z_2-z_0}_{z_1-z_0}$$
My question is what exactly is the operation implied by having multiple ranges (x_2-x_0 to x_1-x_0, y_2-y_0 to y_1-y_0, and z_2-z_0 to z_1-z_0)? Do you perform the difference operation for each range and then add them together?

Edit: Fixed title.

 

[Mark44]

I have the equation
$$G[xy\textup{ln}(z+R)+xz\textup{ln}(y+R)+yz\textup{ln}(x+R)]|^{x_2-x_0}_{x_1-x_0}|^{y_2-y_0}_{y_1-y_0}|^{z_2-z_0}_{z_1-z_0}$$

First off, this isn't an equation (there's no = in it). Is this supposed to be an iterated integral?

My question is what exactly is the operation implied by having multiple ranges (x_2-x_0 to x_1-x_0, y_2-y_0 to y_1-y_0, and z_2-z_0 to z_1-z_0)? Do you perform the difference operation for each range and then add them together?

Edit: Fixed title.

 

[Hypatio]

First off, this isn't an equation (there's no = in it). Is this supposed to be an iterated integral?

It equals some value, N.

It is (part of) the solution to the equation

$$N=G\int_{x_1}^{x_2}\int_{y_1}^{y_2}\int_{z_1}^{z_2}\frac{1}{R}dzdydx$$

I just don't understand the notation of the three bars in the solution.

 

[Char. Limit]

They are bounds. For example...

$$\int_0^3 x^2 dx = \frac{x^3}{3} |_0^3$$

Does that answer your question?

 

[Hypatio]

They are bounds. For example...

$$\int_0^3 x^2 dx = \frac{x^3}{3} |_0^3$$

Does that answer your question?

I know that they are bounds, but what do I do when there is more than one set of bounds? Do I calculate each of the three and then add them together or something different?

 

[Char. Limit]

I really don't know. I wouldn't work all of the integrals at once, then apply all of the bounds. I would work through one integral at a time.

 

[HallsofIvy]

$$f(x,y, z)\left|_{x= a}^b\left|_{y= c}^d\left|_{z= e}^f= f(b, y, z)- f(a, y, z)\left|_{y= c}^dleft|_{z= e}^f$$
$$= f(b, d, z)- f(a, c, z)- (f(b, d, z)- f(b, c, z))\left|_{z= e}^f= f(b, d, z)+ f(b, c, z)- f(a, c, z)- f(b, d, z))\left|_{z= e}^f$$
$$= f(b, d, z)+ f(a, c, f)+ f(a, d, f)+ f(a, c, f)- (f(a, d, f)+ f(a, c, f)+ f(b, d, e)+ f(b, c, f))$$

In other words, it doesn't matter in which order you evaluate x, y, or z.

(This is integrating over a rectangular solid. In more general cases, where the limits on one integral will depend on another variable, of course, the order is important.)