- #31
Muzza
- 689
- 1
Nope, it's not.
Infinite graph with finite area?
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SUMMARY
The discussion centers on the concept of functions that extend infinitely while having a finite area under their curves. Key examples include the error function, defined as \(\frac{1}{2\pi}e^{-x^2}\), and the exponential decay function \(e^{-kx}\) for \(k > 0\). Participants explore the integrals of these functions, particularly \(\int_{-\infty}^{\infty} e^{-x^2}dx\), which equals \(\sqrt{\pi}\), and discuss the implications of Fubini's theorem in evaluating double integrals. The conversation highlights the importance of understanding convergence in improper integrals.
PREREQUISITES- Understanding of improper integrals and convergence
- Familiarity with the error function and its properties
- Knowledge of Fubini's theorem for double integrals
- Basic calculus concepts, including limits and integration techniques
- Study the properties of the error function and its applications in probability
- Learn about the convergence of improper integrals and techniques for evaluation
- Explore Fubini's theorem in depth, particularly its applications in multiple integrals
- Investigate the concept of Cauchy principal value in the context of improper integrals
Students in calculus, mathematicians interested in analysis, and anyone exploring the properties of functions with infinite limits and finite areas under their curves.
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