Expert: Scotto - 1/3/2009

Question
Dear sir/madam
I am doing B.TECH.IT in an engineering college , i find it difficult to solve multiple integrals please say me sites which offer answers for the questions asked so that it would be easy for me.

Answer
The way to do multiple integrals is to treat all of the other variables as constants.
For example, if you have a function f(x,y,z) that needs integrated with respect to x, y, and z, they need to be done one at a time.  To do the integral with respect to x, treat y and z as constants and do x alone.  If there are any limits for x, put in the upper and subtract the lower.  Do the same for y and then for z.

Multiple integrals are just like doing a single integral while holding the other variables constant.  For example, to integrate xyz with respect to x, then y, then z, for each of the varialbes not that the integral is x²/2,y²/2, and z²/2 while holding the other variables constant.  So the entire triple integral is x²y²z²/8, gotten by multiplying all of them together.  I will show you:

⌠⌠⌠
⌡⌡⌡xyz dx dy dz =

⌠⌠
⌡⌡x²yz/2 dy dz =

⌠
⌡x²y²z/(2*2) dz =

x²y²z²/(2*2*2) = x²y²z²/8.


http://tutorial.math.lamar.edu/Classes/CalcIII/MultipleIntegralsIntro.aspx
has definition of Double Integrals,
Iterated Integrals
Double Integrals over General Regions,
Double Integrals in Polar Coordinates,
Triple Integrals,
Triple Integrals in Cylindrical Coordinates,
Triple Integrals in Spherical Coordinates,
Change of Variables,
Surface Area, and
Area and Volume Revisited.
Each of these can be clicked on to get a better understanding.

http://en.wikipedia.org/wiki/Double_integral
has the definition with several examples.

http://ltcconline.net/greenl/courses/202/multipleIntegration/multipleIntegration...
has a list of explanations of
Iterated Integrals and Area
Double Integrals and Volume
Double Integration in Polar Coordinates
Center of Mass and Moment of Inertia
Surface Area
Triple Integrals
Triple Integrals in Cylindrical and Spherical Coordinates
Jacobians

Here’s only three of them.  Now in my words processor that I’m using to input these, the mouse can be used to click on one of the http:://…., but I’m not sure that still holds when this message is transported to AllExperts.  If not, you should be able to use the mouse to highlight them by pushing down on one of the buttons on the mouse.  Once they are highlighted, push cntl-C.  This makes a copy of them.   Go to the address block and type Cntl-V.  This makes a copy of what was just cut.  Note that Cntl-C and Cntl-V work in almost any language I have seen on the computer.

They were around even before networks were introduced to the world back when a PC cost close to $2000 (back in early 1980’s) and didn’t even have a hard disk.