Abstract
A notion of the integral is one of the main ideas of mathematical analysis (or simply “analysis”). In some literature integrals are introduced in two steps: initially, the so-called indefinite integral is defined as an inverse operation to differentiation; then a definite integral is introduced via some natural applications such as an area under a curve. In this chapter we shall use a much clearer and more rigorous route based on defining the definite integral first as a limit of an integral sum. Then the indefinite integral is introduced in a natural way simply as a convenience tool.
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Notes
- 1.
Which is due to Gottfried Leibniz and Jean Baptiste Joseph Fourier.
- 2.
Of course, when D < 0 the parabola appears completely above the horizontal x axis and hence does not have real roots.
- 3.
We shall learn rather soon in Sect. 5.3 when considering functions of many variables that this is called a partial derivative with respect to λ of the function f(x, λ) of two variables.
- 4.
We shall see later on that the derivative of f with respect to λ in Eq. (4.65) should really be denoted as \(\partial f/\partial \lambda\) using the symbol ∂ rather than d, since it is actually a partial derivative of f with respect to its other variable λ as was already mentioned.
- 5.
This is a particular case of one of the special functions, the so-called Gamma function.
- 6.
That is the function is equal to ±∞ at an isolated point between −∞ and + ∞.
- 7.
Compare with Sect. 4.5.4.
- 8.
It is said that it diverges logarithmically.
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Kantorovich, L. (2016). Integral. In: Mathematics for Natural Scientists. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2785-2_4
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