Abstract
Integration naturally follows from the idea of summation, and this route is chosen to introduce the idea of the integral. Integration follows from the concept of an infinite convergent sum, and for this reason it is discussed before differentiation. The continuum limit is taken to show one obtains a well defined limit. Definite and indefinite integrals, as well multiple integrals are defined. The integration of special functions is obtained from first principles. Gaussian integration for single and N-variables are studied. Examples from economics and finance are analyzed.
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Notes
- 1.
The sum of the square of integers is given by
$$\begin{aligned} 1^2+2^2+3^2+\cdots +N^2=\frac{1}{6}N(N+1)(2N+1) \end{aligned}$$.
- 2.
The sum of the first N cubes is given by
$$\begin{aligned} 1^3+2^3+3^3+\cdots +N^3=\frac{1}{4}N^2(N+1)^2 \end{aligned}$$.
- 3.
Of course not all infinite summations are convergent, and so we need to decide what is the criterion of convergence, which will be addressed later.
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Baaquie, B.E. (2020). Integration. In: Mathematical Methods and Quantum Mathematics for Economics and Finance. Springer, Singapore. https://doi.org/10.1007/978-981-15-6611-0_5
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DOI: https://doi.org/10.1007/978-981-15-6611-0_5
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