Exploring Multivariable Calculus: Double Integrals & Beyond

[PrudensOptimus]

OK OK I know Double Integral is from Multivar Calculus,

I was just wondering what we use it for... I heard is good for volumes, but can't yhou also find volumes by just 1 integral?

And also, aside from integrals in Multivar calc, what else are useful?

I want to get a intro to it, can anyone give a lecture or link to a intro page? Thanks.

 

[Jacob Chestnut]

While you're correct that a volume, or area, can be obtained from a single integral, the more general double integral is better suited to more complex problems.

It allows you to integrate a function over an area, while the triple integral does the same over a volume. There are really an infinite number of applications, but a good one is finding mass from density.

Using the double integral one can take the density function and integrate it over an area, thus finding the mass of a lamina. With a triple integral, you can find the mass of any relatively simple solid region, for which a density function exists.

If you'd like to see more about this, I've had good luck here:
http://www.math.hmc.edu/calculus/tutorials/

 

[Ambitwistor]

No, you cannot calculate volumes with just one integral, if you are speaking of the volume of an arbitrary-shaped region. (Highly symmetric regions can be reduced to single integrals, because you can implicitly pull out some of the integrals as constants.)

 

[Jacob Chestnut]

I was referring to the method of finding volume-using areas of revolution or concentric shells. Which of course does only apply to highly symmetric shapes. I assume this is what you're referring to.