FUBINI'S THEOREM: application exercise (double integrals)

Fubini's Theorem is a mathematical theorem that states that, under certain conditions, the order of integration in a multiple integral can be changed. It is named after the Italian mathematician Guido Fubini.
Statement of Fubini's Theorem
Let f(x,y) be a function integrable on a rectangle R = ×. So:

∫∫R f(x,y) dx dy = ∫ (∫ f(x,y) dy) dx = ∫ (∫ f(x,y) dx) dy
Conditions
1. The function f(x,y) must be integrable in R.
2. The integral must be absolutely convergent.
Applications
1. Calculation of volumes and areas.
2. Integration of functions in multidimensional spaces.
3. Fourier analysis.
4. Partial differential equations.
Example
Calculate ∫∫R (x^2 + y^2) dx dy, where R = ×.
Importance
Fubini's Theorem is fundamental in:

1. Mathematical analysis.
2. Integral calculus.
3. Geometry and topology.
4. Physics and engineering.
Generalizations
Fubini's Theorem can be generalized to:

1. Multiple integrals in Euclidean spaces.
2. Lebesgue integrals.
3. Stieltjes integrals.