Khanh: /* Methods */

2021-11-14T05:28:41Z

Methods

Khanh at 05:27, 14 November 2021

2021-11-14T05:27:42Z

Khanh at 17:04, 17 September 2021

2021-09-17T17:04:09Z

Khanh: Created page with "File:Riemann sum convergence.png|right|thumb|300px|Four of the Riemann summation methods for approximating the area under curves. '''Right'''
2021-09-17T17:01:49Z

Created page with "File:Riemann sum convergence.png|right|thumb|300px|Four of the Riemann summation methods for approximating the area under curves. '''Right''' </s..."

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[[File:Riemann sum convergence.png|right|thumb|300px|Four of the Riemann summation methods for approximating the area under curves. '''Right''' and '''left''' methods make the approximation using the right and left endpoints of each subinterval, respectively. '''Maximum''' and '''minimum''' methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively. The values of the sums converge as the subintervals halve from top-left to bottom-right.]]

In mathematics, a '''Riemann sum''' is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations.

The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together. This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution.

Because the region filled by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured. This error can be reduced by dividing up the region more finely, using smaller and smaller shapes. As the shapes get smaller and smaller, the sum approaches the Riemann integral.

==Definition==
Let <math>f: [a,b] \rightarrow \mathbb R</math> be a function defined on a closed interval <math>[a,b]</math> of the real numbers, <math>\mathbb R</math>, and
:<math>P = \left \{[x_0,x_1],[x_1,x_2],\dots,[x_{n-1},x_n] \right \}</math>,
be a partition of ''I'', where
:<math>a=x_0<x_1<x_2<\cdots<x_n=b</math>.
A '''Riemann sum''' <math>S</math> of ''f'' over ''I'' with partition ''P'' is defined as
:<math>S = \sum_{i=1}^{n} f(x_i^)\, \Delta x_i</math>
where <math>\Delta x_i=x_i-x_{i-1}</math> and <math>x_i^\in[x_{i-1},x_i]</math>.
One might produce different Riemann sums depending on which <math>x_i^</math>'s are chosen. In the end this will not matter, if the function is Riemann integrable, when the difference or width of the summands <math>\Delta x_i</math> approaches zero.

==Some specific types of Riemann sums==
Specific choices of <math>x_i^</math> give us different types of Riemann sums:
* If <math>x_i^=x_{i-1}</math> for all ''i'', then ''S'' is called a '''left rule''' or '''left Riemann sum'''.
If <math>x_i^=x_i</math> for all ''i'', then ''S'' is called a '''right rule''' or '''right Riemann sum'''.
If <math>x_i^=(x_i+x_{i-1})/2</math> for all ''i'', then ''S'' is called the '''midpoint rule''' or '''middle Riemann sum'''.
If <math>f(x_i^) = \sup f([x_{i-1},x_i])</math> (that is, the supremum of ''f'' over <math>[x_{i-1},x_i]</math>), then ''S'' is defined to be an '''upper Riemann sum''' or '''upper Darboux sum'''.
If <math>f(x_i^) = \inf f([x_{i-1},x_i])</math> (that is, the infimum of ''f'' over <math>[x_{i-1},x_i]</math>), then ''S'' is defined to be a '''lower Riemann sum''' or '''lower Darboux sum'''.

All these methods are among the most basic ways to accomplish numerical integration. Loosely speaking, a function is Riemann integrable if all Riemann sums converge as the partition "gets finer and finer".

While not derived as a Riemann sum, the average of the left and right Riemann sums is the '''trapezoidal sum''' and is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.

Any Riemann sum on a given partition (that is, for any choice of <math>x_i^</math> between <math>x_{i-1}</math> and <math>x_i</math>) is contained between the lower and upper Darboux sums. This forms the basis of the Darboux integral, which is ultimately equivalent to the Riemann integral.

==Methods==
The four methods of Riemann summation are usually best approached with partitions of equal size. The interval {{Closed-closed|''a'', ''b''}} is therefore divided into <math> n </math> subintervals, each of length

:<math>\Delta x = \frac{b-a}{n}.</math>

The points in the partition will then be

: <math>a, \; a + \Delta x, \; a + 2 \Delta x, \; \ldots, \; a + (n-2) \Delta x, \; a + (n-1)\Delta x, \; b.</math>

===Left Riemann sum===
[[Image:LeftRiemann2.svg|thumb|right|Left Riemann sum of ''x''3 over [0,2] using 4 subdivisions]]
For the left Riemann sum, approximating the function by its value at the left-end point gives multiple rectangles with base Δ''x'' and height ''f''(''a'' + ''i''Δ''x''). Doing this for ''i'' = 0, 1, …, ''n'' − 1, and adding up the resulting areas gives

:<math> A_\mathrm{left} = \Delta x \left[f(a) + f(a + \Delta x) + f(a+ 2\Delta x)+ \cdots+f(b - \Delta x)\right].</math>

The left Riemann sum amounts to an overestimation if ''f'' is monotonically decreasing on this interval, and an underestimation if it is monotonically increasing.

===Right Riemann sum===
[[Image:RightRiemann2.svg|thumb|right|Right Riemann sum of ''x''3 over [0,2] using 4 subdivisions]]
''f'' is here approximated by the value at the right endpoint. This gives multiple rectangles with base Δ''x'' and height ''f''(''a'' + ''i'' Δ''x''). Doing this for ''i'' = 1, …, ''n'', and adding up the resulting areas produces
:<math> A_\mathrm{right} = \Delta x \left[ f( a + \Delta x ) + f(a + 2 \, \Delta x)+\cdots+f(b) \right].</math>

The right Riemann sum amounts to an underestimation if ''f'' is monotonically decreasing, and an overestimation if it is monotonically increasing.
The error of this formula will be

:<math>\left \vert \int_{a}^{b} f(x) \, dx - A_\mathrm{right} \right \vert \le \frac{M_1 (b-a)^2}{2n}</math>,

where <math>M_1</math> is the maximum value of the absolute value of <math>f^{\prime}(x)</math> on the interval.

===Midpoint rule===
[[Image:MidRiemann2.svg|thumb|right|Midpoint Riemann sum of ''x''3 over [0,2] using 4 subdivisions]]
Approximating ''f'' at the midpoint of intervals gives ''f''(''a'' + Δ''x''/2) for the first interval, for the next one ''f''(''a'' + 3Δ''x''/2), and so on until ''f''(''b'' − Δ''x''/2). Summing up the areas gives

:<math> A_\mathrm{mid} = \Delta x\left[f(a + \tfrac{\Delta x}{2}) + f(a + \tfrac{3\,\Delta x}{2}) + \cdots+f(b-\tfrac{\Delta x}{2})\right]</math>.

The error of this formula will be

:<math>\left \vert \int_a^b f(x) \, dx - A_\mathrm{mid} \right \vert \le \frac{M_2(b-a)^3}{24n^2}</math>,

where <math>M_2</math> is the maximum value of the absolute value of <math>f^{\prime\prime}(x)</math> on the interval.

===Trapezoidal rule===
[[Image:TrapRiemann2.svg|thumb|right|Trapezoidal Riemann sum of ''x''3 over [0,2] using 4 subdivisions]]

In this case, the values of the function ''f'' on an interval are approximated by the average of the values at the left and right endpoints. In the same manner as above, a simple calculation using the area formula
:<math>A=\tfrac{1}{2}h(b_1+b_2)</math>
for a trapezium with parallel sides ''b''1, ''b''2 and height ''h'' produces

:<math>A_\mathrm{trap} = \tfrac{1}{2}\,\Delta x\left[f(a) + 2f(a+\Delta x) + 2f(a+2\,\Delta x) +\cdots+f(b)\right].</math>

The error of this formula will be

:<math>\left \vert \int_{a}^{b} f(x) \, dx - A_\mathrm{trap} \right \vert \le \frac{M_2(b-a)^3}{12n^2},</math>

where <math>M_2</math> is the maximum value of the absolute value of <math>f^{\prime\prime}(x)</math>.

The approximation obtained with the trapezoid rule for a function is the same as the average of the left hand and right hand sums of that function.

==Connection with integration==
For a one-dimensional Riemann sum over domain <math>[a,b]</math>, as the maximum size of a partition element shrinks to zero (that is the limit of the norm of the partition goes to zero), some functions will have all Riemann sums converge to the same value. This limiting value, if it exists, is defined as the definite Riemann integral of the function over the domain,

: <math>\int_a^b \! f(x) \, dx = \lim_{\|\Delta x\|\rightarrow0} \sum_{i=1}^{n} f(x_i^) \,\Delta x_i.</math>

For a finite-sized domain, if the maximum size of a partition element shrinks to zero, this implies the number of partition elements goes to infinity. For finite partitions, Riemann sums are always approximations to the limiting value and this approximation gets better as the partition gets finer. The following animations help demonstrate how increasing the number of partitions (while lowering the maximum partition element size) better approximates the "area" under the curve:
<gallery>
Image:Riemann sum (leftbox).gif|Left sum
Image:Riemann sum (rightbox).gif|Right sum
Image:Riemann sum (middlebox).gif|Middle sum
</gallery>
Since the red function here is assumed to be a smooth function, all three Riemann sums will converge to the same value as the number of partitions goes to infinity.

==Example==
[[File:Area-under-curve-for-x-squared.svg|thumb|Area-under-curve-for-x-squared|A visual representation of the area under the curve ''y'' = ''x''2 for the interval from 0 to 2. Using antiderivatives this area is exactly 8/3.]]
[[File:Riemann sum (y=x^2).gif|thumb|Riemann sum (y=x^2)|Approximating the area under <math>y=x^2</math> from 0 to 2 using right-rule sums. Notice that because the function is monotonically increasing, right-hand sums will always overestimate the area contributed by each term in the sum (and do so maximally)]]
[[File:Riemann sum error.svg|thumb|Riemann sum error|The value of the Riemann sum under the curve ''y'' = ''x''2 from 0 to 2. As the number of rectangles increases, it approaches the exact area of 8/3.]]

Taking an example, the area under the curve of ''y'' = ''x''2 between 0 and 2 can be procedurally computed using Riemann's method.

The interval [0, 2] is firstly divided into ''n'' subintervals, each of which is given a width of <math>\tfrac{2}{n}</math>; these are the widths of the Riemann rectangles (hereafter "boxes"). Because the right Riemann sum is to be used, the sequence of ''x'' coordinates for the boxes will be <math>x_1, x_2, \ldots, x_n</math>. Therefore, the sequence of the heights of the boxes will be <math>x_1^2, x_2^2, \ldots, x_n^2</math>. It is an important fact that <math>x_i = \tfrac{2i}{n}</math>, and <math>x_n = 2</math>.

The area of each box will be <math>\tfrac{2}{n} \times x_i^2</math> and therefore the ''n''th right Riemann sum will be:
:<math>\begin{align}
S &= \frac{2}{n} \times \left(\frac{2}{n}\right)^2 + \cdots + \frac{2}{n} \times \left(\frac{2i}{n}\right)^2 + \cdots + \frac{2}{n} \times \left(\frac{2n}{n}\right)^2 \\
&= \frac{8}{n^3} \left(1 + \cdots + i^2 + \cdots + n^2\right)\\
&= \frac{8}{n^3} \left(\frac{n(n+1)(2n+1)}{6}\right)\\
&= \frac{8}{n^3} \left(\frac{2n^3+3n^2+n}{6}\right)\\
&= \frac{8}{3} + \frac{4}{n} + \frac{4}{3n^2}
\end{align}</math>

If the limit is viewed as ''n'' → ∞, it can be concluded that the approximation approaches the actual value of the area under the curve as the number of boxes increases. Hence:
:<math>\lim_{n \to \infty} S = \lim_{n \to \infty}\left(\frac{8}{3} + \frac{4}{n} + \frac{4}{3n^2}\right) = \frac{8}{3}</math>

This method agrees with the definite integral as calculated in more mechanical ways:

:<math>\int_0^2 x^2\, dx = \frac{8}{3}</math>

Because the function is continuous and monotonically increasing on the interval, a right Riemann sum overestimates the integral by the largest amount (while a left Riemann sum would underestimate the integral by the largest amount). This fact, which is intuitively clear from the diagrams, shows how the nature of the function determines how accurate the integral is estimated. While simple, right and left Riemann sums are often less accurate than more advanced techniques of estimating an integral such as the Trapezoidal rule or Simpson's rule.

The example function has an easy-to-find anti-derivative so estimating the integral by Riemann sums is mostly an academic exercise; however it must be remembered that not all functions have anti-derivatives so estimating their integrals by summation is practically important.
{{clear}}

==Higher dimensions==
The basic idea behind a Riemann sum is to "break-up" the domain via a partition into pieces, multiply the "size" of each piece by some value the function takes on that piece, and sum all these products. This can be generalized to allow Riemann sums for functions over domains of more than one dimension.

While intuitively, the process of partitioning the domain is easy to grasp, the technical details of how the domain may be partitioned get much more complicated than the one dimensional case and involves aspects of the geometrical shape of the domain.

===Two dimensions===
In two dimensions, the domain, <math>A</math> may be divided into a number of cells, <math>A_i</math> such that <math display="inline">A = \bigcup_i A_i</math>. In two dimensions, each cell then can be interpreted as having an "area" denoted by <math>\Delta A_i</math>. The Riemann sum is
:<math>S = \sum_{i=1}^n f(x_i^,y_i^)\,\Delta A_i,</math>
where <math>(x_i^,y_i^) \in A_i</math>.

===Three dimensions===
In three dimensions, it is customary to use the letter <math>V</math> for the domain, such that <math display="inline">V = \bigcup_i V_i</math> under the partition and <math>\Delta V_i</math> is the "volume" of the cell indexed by <math>i</math>. The three-dimensional Riemann sum may then be written as

:<math>S = \sum_{i=1}^{n} f(x_i^,y_i^,z_i^) \,\Delta V_i</math>
with <math>(x_i^,y_i^,z_i^) \in V_i</math>.

===Arbitrary number of dimensions===
Higher dimensional Riemann sums follow a similar as from one to two to three dimensions. For an arbitrary dimension, n, a Riemann sum can be written as
:<math>S = \sum_i f(P_i^)\, \Delta V_i</math>
where <math>P_i^\in V_i</math>, that is, it's a point in the n-dimensional cell <math>V_i</math> with n-dimensional volume <math>\Delta V_i</math>.

===Generalization===
In high generality, Riemann sums can be written
:<math>S = \sum_i f(P_i^) \mu(V_i)</math>
where <math>P_i^</math> stands for any arbitrary point contained in the partition element <math>V_i</math> and <math>\mu</math> is a measure on the underlying set. Roughly speaking, a measure is a function that gives a "size" of a set, in this case the size of the set <math>V_i</math>; in one dimension, this can often be interpreted as the length of the interval, in two dimensions, an area, in three dimensions, a volume, and so on.

==Resources==
* [https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/a/left-and-right-riemann-sums|Left & Right Riemann Sums Article], Khan Academy
* [https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/v/simple-riemann-approximation-using-rectangles|Riemann approximation introduction], Khan Academy
* [https://www.youtube.com/watch?v=AkUa9Fkz2rw|Riemann Sums - Midpoint, Left & Right Endpoints, Area, Definite Integral, Sigma Notation, Calculus], The Organic Chemistry Tutor

==References==
# Hughes-Hallett, Deborah; McCullum, William G.; et al. (2005). Calculus (4th ed.). Wiley. p. 252. (Among many equivalent variations on the definition, this reference closely resembles the one given here.)
# Hughes-Hallett, Deborah; McCullum, William G.; et al. (2005). Calculus (4th ed.). Wiley. p. 340. So far, we have three ways of estimating an integral using a Riemann sum: 1. The left rule uses the left endpoint of each subinterval. 2. The right rule uses the right endpoint of each subinterval. 3. The midpoint rule uses the midpoint of each subinterval.
# Ostebee, Arnold; Zorn, Paul (2002). Calculus from Graphical, Numerical, and Symbolic Points of View (Second ed.). p. M-33. Left-rule, right-rule, and midpoint-rule approximating sums all fit this definition.
# Swokowski, Earl W. (1979). Calculus with Analytic Geometry (Second ed.). Boston, MA: Prindle, Weber & Schmidt. pp. 821–822. ISBN 0-87150-268-2.
# Ostebee, Arnold; Zorn, Paul (2002). Calculus from Graphical, Numerical, and Symbolic Points of View (Second ed.). p. M-34. We chop the plane region R into m smaller regions R1, R2, R3, ..., Rm, perhaps of different sizes and shapes. The 'size' of a subregion Ri is now taken to be its area, denoted by ΔAi.
# Swokowski, Earl W. (1979). Calculus with Analytic Geometry (Second ed.). Boston, MA: Prindle, Weber & Schmidt. pp. 857–858. ISBN 0-87150-268-2.

@@ Line 164: / Line 164: @@
 * [https://www.youtube.com/watch?v=AkUa9Fkz2rw Riemann Sums - Midpoint, Left & Right Endpoints, Area, Definite Integral, Sigma Notation, Calculus], The Organic Chemistry Tutor
-==References==
+== Licensing ==
-# Hughes-Hallett, Deborah; McCullum, William G.; et al. (2005). Calculus (4th ed.). Wiley. p. 252. (Among many equivalent variations on the definition, this reference closely resembles the one given here.)
+Content obtained and/or adapted from:
-# Hughes-Hallett, Deborah; McCullum, William G.; et al. (2005). Calculus (4th ed.). Wiley. p. 340. So far, we have three ways of estimating an integral using a Riemann sum: 1. The left rule uses the left endpoint of each subinterval. 2. The right rule uses the right endpoint of each subinterval. 3. The midpoint rule uses the midpoint of each subinterval.
+* [https://en.wikipedia.org/wiki/Riemann_sum Riemann sum, Wikipedia] under a CC BY-SA license

@@ Line 32: / Line 32: @@
 ==Methods==
-The four methods of Riemann summation are usually best approached with partitions of equal size. The interval {{Closed-closed|''a'', ''b''}} is therefore divided into <math> n </math> subintervals, each of length
+The four methods of Riemann summation are usually best approached with partitions of equal size. The interval [''a'', ''b''] is therefore divided into <math> n </math> subintervals, each of length
 :<math>\Delta x = \frac{b-a}{n}.</math>

@@ Line 131: / Line 131: @@
 The example function has an easy-to-find anti-derivative so estimating the integral by Riemann sums is mostly an academic exercise; however it must be remembered that not all functions have anti-derivatives so estimating their integrals by summation is practically important.
-{{clear}}
 ==Higher dimensions==
@@ Line 160: / Line 160: @@
 ==Resources==
-* [https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/a/left-and-right-riemann-sums|Left & Right Riemann Sums Article], Khan Academy
+* [https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/a/left-and-right-riemann-sums Left & Right Riemann Sums Article], Khan Academy
-* [https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/v/simple-riemann-approximation-using-rectangles|Riemann approximation introduction], Khan Academy
+* [https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/v/simple-riemann-approximation-using-rectangles Riemann approximation introduction], Khan Academy
-* [https://www.youtube.com/watch?v=AkUa9Fkz2rw|Riemann Sums - Midpoint, Left & Right Endpoints, Area, Definite Integral, Sigma Notation, Calculus], The Organic Chemistry Tutor
+* [https://www.youtube.com/watch?v=AkUa9Fkz2rw Riemann Sums - Midpoint, Left & Right Endpoints, Area, Definite Integral, Sigma Notation, Calculus], The Organic Chemistry Tutor
 ==References==

Area of a rectangle - Revision history

Khanh: /* Methods */

Khanh at 05:27, 14 November 2021

Khanh at 17:04, 17 September 2021