Polymer physics

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 an approximation of the area of a region, often the region underneath a curve. It is named after German mathematician Bernhard Riemann.

The sum is calculated by dividing the region up into shapes (rectangles or trapezoids) 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 f : D → R be a function defined on a subset, D, of the real line, R. Let I = [a, b] be a closed interval contained in D, and let

$P=\left\{[x_{0},x_{1}),[x_{1},x_{2}),\dots ,[x_{n-1},x_{n}]\right\},$

be a partition of I, where

$a=x_{0}<x_{1}<x_{2}<\cdots <x_{n}=b.$

The Riemann sum of f over I with partition P is defined as

$S=\sum _{i=1}^{n}f(x_{i}^{*})(x_{i}-x_{i-1}),\quad x_{i-1}\leq x_{i}^{*}\leq x_{i}.$

The choice of $x_{i}^{*}$ in the interval $[x_{i-1},x_{i}]$ is arbitrary.

Example: Specific choices of $x_{i}^{*}$ give us different types of Riemann sums:

• If $x_{i}^{*}=x_{i-1}$ for all i, then S is called a left Riemann sum.
• If $x_{i}^{*}=x_{i}$ for all i, then S is called a right Riemann sum.
• If $x_{i}^{*}={\tfrac {1}{2}}(x_{i}+x_{i-1})$ for all i, then S is called a middle Riemann sum.
• The average of the left and right Riemann sum is the trapezoidal sum.
• If it is given that

$S=\sum _{i=1}^{n}v_{i}(x_{i}-x_{i-1}),$

where $v_{i}$ is the supremum of f over $[x_{i-1},x_{i}]$, then S is defined to be an upper Riemann sum.

• Similarly, if $v_{i}$ is the infimum of f over $[x_{i-1},x_{i}]$, then S is a lower Riemann sum.

People are nonetheless rushing to buy properties in Singapore. A new residential condominium mission in Bedok had a long queue three days before its official launch and bought more than 350 units on the very first day. Nonetheless, there is a ‘development' that had emerged in 2011 that the majority have possibly overlooked.

In August, more than a hundred individuals across 16 states reportedly had been sickened with salmonella after consuming mangos In September, the Meals and Drug Administration detained mango imports from a Mexican packing house after the fruits had been linked to the sickenings Save Cash On Your Online Purchasing Make saving money when purchasing on-line a breeze with Bodge's favourite app, Invisible Hands. This helpful program does the price-checking analysis for you, and will warn you when a unique site has a better value - or let you realize that you have found the most effective deal. Watch Posted by Edison Foo September 29, 2013 HEARTH SALE!!! From only $9xx,000! We hope this listing will help launch your search for new properties in Singapore. It's unlawful to purchase ice cream after 6 pm, in Newark.

for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed

After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant

Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013

6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!

BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill Any Riemann sum on a given partition (that is, for any choice of $x_{i}^{*}$ between $x_{i-1}$ and $x_{i}$) is contained between the lower and the upper Riemann sums. A function is defined to be Riemann integrable if the lower and upper Riemann sums get ever closer as the partition gets finer and finer. This fact can also be used for numerical integration.

Methods

The four methods of Riemann summation are usually best approached with partitions of equal size. The interval [a, b] is therefore divided into n subintervals, each of length

$\Delta x={\frac {b-a}{n}}.$

The points in the partition will then be

$a,a+\Delta x,a+2\Delta x,\ldots ,a+(n-2)\Delta x,a+(n-1)\Delta x,b.$

Left Riemann Sum

For the left Riemann sum, approximating the function by its value at the left-end point gives multiple rectangles with base Δx and height f(a + iΔx). Doing this for i = 0, 1, ..., n − 1, and adding up the resulting areas gives

$\Delta x\left[f(a)+f(a+\Delta x)+f(a+2\Delta x)+\cdots +f(b-\Delta x)\right].$

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

f is here approximated by the value at the right endpoint. This gives multiple rectangles with base Δx and height f(a + iΔx). Doing this for i = 1, ..., n, and adding up the resulting areas produces

$\Delta x\left[f(a+\Delta x)+f(a+2\Delta x)+\cdots +f(b)\right].$

The right Riemann sum amounts to an underestimation if f is monotonically decreasing, and an overestimation if it is monotonically increasing.

Middle sum

Approximating f at the midpoint of intervals gives f(a + Q/2) for the first interval, for the next one f(a + 3Q/2), and so on until f(b − Q/2). Summing up the areas gives

$Q\left[f(a+{\tfrac {Q}{2}})+f(a+{\tfrac {3Q}{2}})+\cdots +f(b-{\tfrac {Q}{2}})\right].$

The error of this formula will be

$\left\vert \int _{a}^{b}f(x)\,dx-A_{\mathrm {mid} }\right\vert \leq {\frac {M_{2}(b-a)^{3}}{24n^{2}}},$

where $M_{2}$ is the maximum value of the absolute value of $f^{\prime \prime }(x)$ on the interval.

Trapezoidal Rule

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

$A={\tfrac {1}{2}}h(b_{1}+b_{2})$

for a trapezium with parallel sides b₁, b₂ and height h produces

${\tfrac {1}{2}}Q\left[f(a)+2f(a+Q)+2f(a+2Q)+2f(a+3Q)+\cdots +f(b)\right].$

The error of this formula will be

$\left\vert \int _{a}^{b}f(x)\,dx-A_{\mathrm {trap} }\right\vert \leq {\frac {M_{2}(b-a)^{3}}{12n^{2}}},$

where $M_{2}$ is the maximum value of the absolute value of $f^{\prime \prime }(x).$

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.

Example

A visual representation of the area under the curve y = x² for the interval from 0 to 2

The value of the Riemann sum under the curve y = x² 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 = x² 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 ${\tfrac {2}{n}}$; 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 $x_{1},x_{2},\ldots ,x_{n}$. Therefore, the sequence of the heights of the boxes will be $x_{1}^{2},x_{2}^{2},\ldots ,x_{n}^{2}$. It is an important fact that $x_{i}={\tfrac {2i}{n}}$, and $x_{n}=2$.

The area of each box will be ${\tfrac {2}{n}}\times x_{i}^{2}$ and therefore the nth right Riemann sum will be:

${\begin{aligned}S&={\frac {2}{n}}\times \left({\frac {2}{n}}\right)^{2}+\cdots +{\frac {2}{n}}\times \left({\frac {2i}{n}}\right)^{2}+\cdots +{\frac {2}{n}}\times \left({\frac {2n}{n}}\right)^{2}\\&={\frac {8}{n^{3}}}\left(1+\cdots +i^{2}+\cdots +n^{2}\right)\\&={\frac {8}{n^{3}}}\left({\frac {n(n+1)(2n+1)}{6}}\right)\\&={\frac {8}{n^{3}}}\left({\frac {2n^{3}+3n^{2}+n}{6}}\right)\\&={\frac {8}{3}}+{\frac {4}{n}}+{\frac {4}{3n^{2}}}\end{aligned}}$

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:

$\lim _{n\to \infty }S=\lim _{n\to \infty }\left({\frac {8}{3}}+{\frac {4}{n}}+{\frac {4}{3n^{2}}}\right)={\frac {8}{3}}$

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

$\int _{0}^{2}x^{2}\,dx={\frac {8}{3}}$

Animations

• 

  Left sum

• 

  Right sum

• 

  Middle sum

• 

  With $y=x^{2}$

See also

• Riemann integral
• Riemann–Stieltjes integral
• Lebesgue integral
• Simpson's rule
• Euler method and midpoint method, related methods for solving differential equations

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• A simulation showing the convergence of Riemann sums