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Is integral calculus and mathematical analysis in a core concept. Usually divided into definite integral and indefinite integral two kinds.
1 Basic definitions
1.1 definite integral
For a given positive real function \ (F (X) \) , \ (F (X) \) in a real interval \ ([a, b] \ ) definite integral on
\ [\ int_a ^ bf ( x) \, dx \]
It will be appreciated on the value of the \ (O_ {xy} \) on the coordinate plane, by curve \ ((x, f (x )) (x \ in [a, b]) \) , straight \ (X = a \) , \ (X = B \) and the X-axis value of the area surrounded by the curved side trapezoid (a method of determining the real value).
Wherein \ (\ mathrm {d} x \) is called the integration variable representing the required range area is calculated using a scale of the abscissa axis; \ (\ int_a ^ b \) indicates start counting from a, b to So far, it referred to the range of integration or integration domain , wherein a is called the integral lower bound , b is called the upper bound integral , \ (\ int \) is called the integral sign , is elongated from the letter S (summa (summa) Latin: summing initials) of evolution. Function \ (f (x) \) written in the middle, called the integrand .
1.2 indefinite integral
\ (f (x) \) indefinite integral (or primary function) refers to any derivative of a function satisfies \ (f (x) \) function \ (F. (X) \) . A function \ (f (x) \) indefinite integral is not unique: as long as the \ (F (x) \) is \ (f (x) \) indefinite integral, then with a constant phase difference function \ (F (x) + C \) is \ (f (x) \) indefinite integral.
The absence of description, the term "integral" hereinafter means "definite integral" .
1.3 Riemann integral
In real analysis, the Riemann founded by the Riemann integral (English: Riemann integral) for the first time at a function on a given interval of integration gives a precise definition. Riemann integral by the technical shortcomings of some later Riemann - Jess Steele integral and Lebesgue integral patched.
The basic concept is to Riemann integral X - axis more finely divided, then it corresponds to a rectangular area will be more and approach pattern S area (see below).
1.3.1 Segmentation section
A closed interval \ ([a, b] \ ) a segmented \ (P \) refers to take a finite sequence of points in this interval \ (a = x_ {0} <x_ {1} <x_ {2} <\ ldots <} = X_ {n-B \) . Each closed interval \ ([x_ {i}, x_ {i + 1}] \) is called a subinterval . Defined \ (\ the lambda \) the maximum length of subintervals: \ (\ the lambda = \ max (X_ {I} -x_ +. 1 {I}) \) , where \ (0 \ leq i \ leq n- 1 \) .
A closed interval \ ([a, b] \ ) of one sample dividing means after performing segmentation, to remove a little in each subinterval \ (x_ {i} \ leq t_ {i} \ leq x_ {i + 1 } \) .
1.3.2 Riemann and
To a closed interval \ ([a, b] \ ) has defined real-valued function \ (F \) , \ (F \) on the sample divided \ (x_ {0}, \ cdots, x_ {n} \) , \ (T_ {0}, \ cdots,. 1-n-T_ {} \) of the Riemann sum (integral sum) and defined by the following formula:
\ [\ Sum_ {i = 0
} ^ {n-1} f (t_ {i}) (x_ {i + 1} -x_ {i}) \] , and wherein each sub-interval length is \ ( x_ {i + 1} -x_ { i} \) and in \ (t_ {i} \) function value at \ (f (t_ {i} ) \) product. Intuitively, it is to mark the point \ (t_ {i} \) distance to the X-axis is high, divided in subranges long rectangular area.