Double Integral and Triple Integral question

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February 2nd 2010, 08:10 AM
nunos

Double Integral and Triple Integral question

So I understand the meaning of an integral of a function, which is the area of the region under the curve that represents that function within the limits of integration.

I also understand the meaning of $\iint_D dxdy$ which is the area of the region D and $\iiint_V dxdydz$ which is the volume of a solid V.

What I don't understand, even though I do it all the time, is the meaning of a double integral (and a triple integral) of a function. For example: $\iint_D (x^2+\sin y)dxdy$ and $\iiint (x^2+\sin y \cos z)dxdydz$ .

Thanks in advance.
February 2nd 2010, 11:18 PM
nunos

Anyone?
February 3rd 2010, 05:56 AM
HallsofIvy

First, you need to unlearn what you think you understand. "Area under the curve" is NOT the "meaning" of the integral, it is one possible application. Nor are "surface area" and "volume" the "meanings" of the double and triple integrals. They are specific problems for which the integral can be used. An integral, like any mathematical object, has no "meaning", as you are using the word, apart from a specific application.

For something like $\int\int f(x,y)dydx$ you can think of f(x,y) as being the height of some object above each point (x, y) in the plane. In that way the integral is the volume of the object. But you could also have f(x,y) as the density of a two-dimensional object. Then the integral would be the total mass of the object. You could also have f(x,y) as the pressure of a liquid or gas on a plane. Then the integral gives the total force on that plane.

For $\int\int\int f(x,y,z)dxdydz$ , if f(x,y,z) is the density of the object, then the integral is total mass. Or it could be the charge density so that the integral is the total charge.
February 3rd 2010, 12:47 PM
nunos

Quote:

Originally Posted by HallsofIvy

First, you need to unlearn what you think you understand. "Area under the curve" is NOT the "meaning" of the integral, it is one possible application. Nor are "surface area" and "volume" the "meanings" of the double and triple integrals. They are specific problems for which the integral can be used. An integral, like any mathematical object, has no "meaning", as you are using the word, apart from a specific application.

For something like $\int\int f(x,y)dydx$ you can think of f(x,y) as being the height of some object above each point (x, y) in the plane. In that way the integral is the volume of the object. But you could also have f(x,y) as the density of a two-dimensional object. Then the integral would be the total mass of the object. You could also have f(x,y) as the pressure of a liquid or gas on a plane. Then the integral gives the total force on that plane.

For $\int\int\int f(x,y,z)dxdydz$ , if f(x,y,z) is the density of the object, then the integral is total mass. Or it could be the charge density so that the integral is the total charge.

Thanks for your reply. That does make sense, but in my Calculus class, I have never talked about density and calculating the mass of objects and yet I calculate such integrals quite often. Am I calculating masses and densities without knowing so?