What's the difference btw double and triple integrals

[denian]

i read both double and triple integrals can be used to find the volume.
so, what's the difference?

 

[chroot]

Double integrals integrate over two variables -- for example, x and y on a plane -- and can be used to calculate areas, but not volumes.

Triple integrals integrate over three variables -- for example, x, y, and z in Cartesian three-dimensional space -- and can be used to calculate volumes.

- Warren

 

[matt grime]

A double integral involves two variables you integrate, a triple 3 variables (guess the pattern...). They can be used to find many things, depending on how you use them, if you use them to 'find' anything.

 

[matt grime]

Sorry, chroot but that isn't exactly accurate.

Take single integrals. They can be used to find path length or areas, same with double and triple integrals. It depends on what the integrand is and what integral is with respect to.

 

[arildno]

Just to illustrate the situation:
Let the volume of a region be given by:
$$V=\iint_{A}\int^{s(x,y)}_{h(x,y)}dzdydx, (x,y)\in{A}$$
Then, we have the trivial conversion into a double integral computation:
$$V=\iint_{A}(s(x,y)-h(x,y))dydx$$

 

[chroot]

Sorry, chroot but that isn't exactly accurate.

Take single integrals. They can be used to find path length or areas, same with double and triple integrals. It depends on what the integrand is and what integral is with respect to.

I know it's not exactly accurate; I suppose I should have said more.

- Warren

 

[quantumdude]

Take single integrals. They can be used to find path length or areas...(snip)

...or even volumes.

i read both double and triple integrals can be used to find the volume.
so, what's the difference?

Long story short: You can use a triple integral with a unit integrand ($\int\int\int dV$) to find the volume of any closed figure in $\mathbb{R}^3$. The boundary of the region is encoded in the limits of integration. You can use double integrals of the form $\int\int f(x_1,x_2) dS$ to find the volume of a region in $\mathbb{R}^3$ provided that the region is partially bounded by the surface $x_3=f(x_1,x_2)$. And lastly you can use single integrals to find volumes of regions of $\mathbb{R}^3$ bounded by surfaces of revolution.

Loosely speaking the fewer integrations in your volume computation, the more picky you have to be about the region.

 

[denian]

tq.

i want to ask another question:

is it possible to find the volume of the solid by double integration,
if only one equation is given

x/3 + y/4 + z/5 = 1

 

[denian]

i got the answer -30
is it correct?