Partition of an interval

Partition of an interval
Tag	`069P20KYQT`
URI	`m2://069P20KYQT`
Kind	definition
Writing	W0 stub
MSC	26A42
Defines	Partition of an interval

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Definition

Let $a < b$ be real numbers. A partition of $[a, b]$ is a finite sequence of real numbers $P = (x_0, x_1, \dots, x_n)$ with $a = x_0 < x_1 < \cdots < x_n = b.$ The intervals $[x_{i-1}, x_i]$ for $i = 1, \dots, n$ are the subintervals of $P$ .

Remarks

Partitions are the basis of Darboux sums and Riemann sums. A refinement of $P$ adds more points. The mesh of $P$ is the length of its longest subinterval. A tagged partition also chooses a point $t_i$ in each subinterval $[x_{i-1}, x_i]$ .

Explanations

L1 human

A partition cuts the interval $[a, b]$ into finitely many pieces. The cut points are listed from left to right.

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What depends on this

55 nodes use this node, directly or through other nodes.

Euler–Mascheroni constant 00C5P9JZBK
Reduction formula 012RYET5E2
Riemann's criterion for integrability 017DH8WXTE
Integral test 019QZ6RRVH
Definite integral 021WEM680W
Darboux and Riemann integrals agree 021XJAQVDC
Integral of exp(-x) over the half line 039JAF7ZAN
Absolutely convergent improper integral 03GP7KMBYM
The Dirichlet function is not Riemann integrable 03Z4JC4YRJ
Mesh of a partition 04GM4NYXJB
Integral of cosine 04WTBQ7APQ
Monotone functions are Riemann integrable 04YKAYDQPW
Lebesgue's criterion for Riemann integrability 0565EGK6XP
Additivity of the integral over intervals 05RCDKHZM6
Harmonic numbers grow like log n 05S2GSPMGA
First fundamental theorem of calculus 063H5KJC69
Comparison of a series and an integral 069HKV4X1E
Integral of the exponential function 06D0EWR2DV
Continuous functions are Riemann integrable 06SESV7GQY
Linearity of the integral 080RWVVSB6
Integration by parts 08Y1BBQTK9
Upper and lower Darboux integrals 09FQT057T7
Uniform limits commute with integration 0BWGW68B66
Integration by substitution 0D37QFPEZE
Gaussian integral 0DFT1GP1JP
Trigonometric substitution 0EAR3KNRTN
Refinement of a partition 0EEKF7YFBW
The gamma function extends the factorial 0FDCW77H44
Comparison test for improper integrals 0FR1BDKK10
Area under a curve 0GXA8Z205E
Integral of sine 0GXCXCX3QN
Dirichlet integral 0GZNSHMG4Q
Riemann sum 0HBY48G8G1
Integral of 1/x 0K1N1Y4EGZ
Stirling's formula 0KC0F51T6A
Step function 0M9XEDF1HJ
Monotonicity of the integral 0MBWQKM8QR
Riemann integrable function 0N1JQXZXHC
Mean value theorem for integrals 0N91N98EG5
The integral of a continuous function is an antiderivative 0NC2A3HWGT
Fundamental theorem of calculus 0NHGY0N0JB
Triangle inequality for integrals 0NR82KMYKD
Integral of a power 0Q3HYFT9RQ
Riemann integral 0QM4TXT3DR
Upper and lower Darboux sums 0QRQVH0CRW
Gamma function 0V3FZMVB4K
Improper integral 0VCTBJMEJF
Improper integral of x^p on an unbounded interval 0VGKB2NC4T
Tangent half-angle substitution 0W8ETY2AZ6
Integration by partial fractions 0WBTWFXJQG
Taylor's theorem with the integral remainder 0WJGR81HBD
Arc length 0X7G61297X
Wallis product 0XCYA4YERS
Second fundamental theorem of calculus 0XYAFCN0PT
Term-by-term integration 0YBY3VYX0K

Sources

Web https://en.wikipedia.org/wiki/Partition_of_an_interval, lead section

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Used by

Mesh of a partition 04GM4NYXJB
Refinement of a partition 0EEKF7YFBW
Riemann sum 0HBY48G8G1
Step function 0M9XEDF1HJ
Upper and lower Darboux sums 0QRQVH0CRW
Arc length 0X7G61297X