Definite integral

A definite integral is an additive monotone functional defined on a set of pairs, the first component of which is an integrable function or functional , and the second is a domain in the set of the task of this function (functional) ^[1] .

Content

Definition

Let be $f(x)$ ${\ displaystyle f (x)}$ defined on the segment $[a;b]$ ${\ displaystyle [a; b]}$ . We will break $[a;b]$ ${\ displaystyle [a; b]}$ to pieces by several arbitrary points: $a=x_{0}<x_{1}<x_{2}<\ldots <x_{n}=b$ ${\ displaystyle a = x_ {0} <x_ {1} <x_ {2} <\ ldots <x_ {n} = b}$ .

Then they say that the partition is done $R$ ${\ displaystyle R}$ segment $[a;b].$ ${\ displaystyle [a; b].}$ Next, choose an arbitrary point $\xi _{i}\in [x_{i};x_{i+1}]$ ${\ displaystyle \ xi _ {i} \ in [x_ {i}; x_ {i + 1}]}$ , $i={\overline {0,n-1}}$ ${\ displaystyle i = {\ overline {0, n-1}}}$ .

A definite integral of a function $f(x)$ ${\ displaystyle f (x)}$ on the segment $[a;b]$ ${\ displaystyle [a; b]}$ the limit of integral sums is called

partitions to zero $\lambda _{R}\rightarrow 0$ ${\ displaystyle \ lambda _ {R} \ rightarrow 0}$ if it exists independently of the partition $R$ ${\ displaystyle R}$ and select points $\xi _{i}$ ${\ displaystyle \ xi _ {i}}$ , i.e

\int \limits _{a}^{b}f(x)dx=\lim \limits _{\Delta x\rightarrow 0}\sum \limits _{i=0}^{n-1}f(\xi _{i})\Delta x_{i}

{\ displaystyle \ int \ limits _ {a} ^ {b} f (x) dx = \ lim \ limits _ {\ Delta x \ rightarrow 0} \ sum \ limits _ {i = 0} ^ {n-1} f (\ xi _ {i}) \ Delta x_ {i}}

If the specified limit exists, then the function $f(x)$ ${\ displaystyle f (x)}$ called integrable on $[a;b]$ ${\ displaystyle [a; b]}$ according to Riemann.

Conventions

$\int \limits _{a}^{b}f(x)dx$ ${\ displaystyle \ int \ limits _ {a} ^ {b} f (x) dx}$

$a$ ${\ displaystyle a}$ - lower limit.
$b$ ${\ displaystyle b}$ - upper limit.
$f(x)$ ${\ displaystyle f (x)}$ - integrand function.
$\Delta x_{i}$ ${\ displaystyle \ Delta x_ {i}}$ - the length of the partial segment.
$\sigma _{R}$ ${\ displaystyle \ sigma _ {R}}$ - integral sum of function $f(x)$ ${\ displaystyle f (x)}$ on $[a;b]$ ${\ displaystyle [a; b]}$ corresponding to the partition $R$ ${\ displaystyle R}$ .
$\lambda _{R}=\sup {\Delta x_{i}}$ ${\ displaystyle \ lambda _ {R} = \ sup {\ Delta x_ {i}}}$ - the maximum of the lengths of partial segments.

Properties

If the function $f(x)$ ${\ displaystyle f (x)}$ Riemann integrable on $[a;b]$ ${\ displaystyle [a; b]}$ then it is limited to it.

Geometric meaning

Defined integral as the area of a figure

Definite integral $\int \limits _{a}^{b}f(x)\,dx$ ${\ displaystyle \ int \ limits _ {a} ^ {b} f (x) \, dx}$ numerically equal to the area of the figure bounded by the abscissa axis, straight $x=a$ ${\ displaystyle x = a}$ and $x=b$ ${\ displaystyle x = b}$ and schedule functions $f(x)$ ${\ displaystyle f (x)}$ .

Calculation Examples

The following are examples of taking certain integrals using the Newton - Leibniz formula .

$\int \limits _{8}^{9}x^{2}\,dx={\frac {x^{3}}{3}}{\Big |}_{8}^{9}={\frac {729}{3}}-{\frac {512}{3}}={\frac {217}{3}}=72{,}(3)\approx 72{,}3$ ${\ displaystyle \ int \ limits _ {8} ^ {9} x ^ {2} \, dx = {\ frac {x ^ {3}} {3}} {\ Big |} _ {8} ^ {9 } = {\ frac {729} {3}} - {\ frac {512} {3}} = {\ frac {217} {3}} = 72 {,} (3) \ approx 72 {,} 3}$
$\int \limits _{1}^{b}{\frac {dx}{x}}=\ln x{\Big |}_{1}^{b}=\ln b$ ${\ displaystyle \ int \ limits _ {1} ^ {b} {\ frac {dx} {x}} = \ ln x {\ Big |} _ {1} ^ {b} = \ ln b}$
$\int \limits _{1}^{4}{\frac {2dx}{x}}=2\ln x{\Big |}_{1}^{4}\approx 2{,}8$ ${\ displaystyle \ int \ limits _ {1} ^ {4} {\ frac {2dx} {x}} = 2 \ ln x {\ Big |} _ {1} ^ {4} \ approx 2 {,} 8 }$

Notes

↑ Great Russian Encyclopedia : [in 35 vols.] / Ch. ed. Yu.S. Osipov . - M .: Great Russian Encyclopedia, 2004—2017.

Literature

Integral - an article from the Great Soviet Encyclopedia .

[БРЭ-1] Great Russian Encyclopedia : [in 35 vols.] / Ch. ed. Yu.S. Osipov . - M .: Great Russian Encyclopedia, 2004—2017.