Integral Theory, Fubini's Theorem

regarding the theoretical definition of double integrals, is it correct to think of them as a kind of double Riemann sums, sums of 2d planes? sometimes the order of integration results in simplification of the calculuation....

once you have mastered Lebegue Integration and measure theory is Fubini's theorem and Riemann integration not useful...

I(limit a to b) I(limit c to d) f(x,y)dydx= I (limit c to d) I (limit a to b) f(x,y)dxdy

thanks very much!

Re: Integral Theory, Fubini's Theorem

I'm not sure I like the phrase "sums of 2D planes"- you can't add planes, they aren't numbers! You can think of $\displaystyle \int_c^d\int_a^b f(x,y)dxdy$ as the volume under the surface z= f(x, y) and the rectangle in the xy-plane, a< x< b, c< y< d. Using Riemann sums, we divide that region into tall thin rectangular solids, having base $\displaystyle \Delta x\Delta y$ and height $\displaystyle f(x^*, y^*)$ where $\displaystyle (x^*, y^*)$ is some point inside one of the small rectangles. Yes, because, at each step, we have only a finite number of such rectangles, it does not matter whether we add the volume "horizontally first, then vertically" or "vertically first, then horizontally". That is the "meat" or Fubini's theorem. All the "hard" work is hidden in the limit process! And we know that works precisely because the function is "integrable".

Re: Integral Theory, Fubini's Theorem

thanks very much! yes it is a double Riemann sum with the number of rectangles becoming infinitely small as n goes to infinity

so for the surface integral are you approximating a surface with a polyhedron with n faces, but what is the meaning of the integrand F(x,y,z) dot n vector delta S? wikipedia describes a parameterization of the surface by applying a new coordinate system on the surface. the dot product is the component of n along F? I'm looking at Schey's Div, Grad, Curl and Griffith's Electrodynamics

Surface integral - Wikipedia, the free encyclopedia

Re: Integral Theory, Fubini's Theorem

by the way is my understanding of the integral correct?

do you mean that Riemann sums can be represented by an nxn matrix and it doesn't matter if you sum the rows or columns first?

with Cavalieri's principle you can calculate volumes of surfaces using the A(x) area of a family of planes Px

Re: Integral Theory, Fubini's Theorem

it seems like a minor point but in a matrix you are adding the area of plane sides of a rectangular box