OpenLLM-France/wikimedia huggingface dataset:
# Physics and Astronomy Labs/Upcoming Test (a)
## For the next Astronomy test on March 18, 2015

Questions not on this list will be on the next astronomy test

## partition for testmakers

questions under constructin that will NOT be on the next test :
Not on Astronomy test
group 1
1
The units for qV=energy
energy
Joule/Coloumb
Joules
Coloumb
2
If an electron's charge equals 2 in some units, and if r is the number of electrons per second, then the the current, I, equals
I=2r
I=(1/2)r
I=4r
I=r
3
If an electron's charge equals 1/2 in some units, and if r is the number of electrons per second, then the the current, I, equals
I=(1/2)r
I=2r
I=4r
I=r
4
In the gravitational analogy to electrostatics and electricity, what is the analog to charge? (Here, m is mass, h is height, and g is gravitational acceleration. Also, the formula for gravitational potential energy is mgh)
m
mg
mgh
gh
Group 2
1
Voltage is expressed as?
energy/q
q/energy
I/gh
gh/q
2
Is current equal to the number of electrons per second flowing through a wire?
only in units where the charge of the electron equals one
yes
no
only in units where the charge of the electron equals the potential energy of a mouse
3
In the gravitational analogy to electrostatics and electricity, what is analogous to voltage? (Here, m is mass, h is height, and g is gravitational acceleration.
m
mg
mgh
gh
4
question?
yes
wrong
wrong
wrong
5
question?
yes
wrong
wrong
wrong :
# Physics and Astronomy Labs/Vernier force probe

We have been having problems with our force measure probes because they are not registering on the lab quest.
We finally figured out how to get the force probes working by hitting cancel on the initial screen on the Lab Quest then plugging in the force probes, but the two force probes gave different reading.
you touch on screen then a menu pops up then click on zero
# Physics and Astronomy Labs/Parallax
## Purpose and design

The purpose of this lab is to use parallax to measure the length of a hallway. Four ring stands are required. These are depicted in the movie shown below, although the movie does not depict the use of these stands to calculate parallax. Instead the movie documents the fact that the distance between the stands must be measured at the top, where the eye is used to line the stands up to a distant object. The figure to the right allows one to derive:
    ${\frac {b}{x-a}}={\frac {c-b}{a}}$
The lab consists of measuring a, b, c, and solving this equation for x.

## Results

If a, b, and c are roughly 2 meters, and the distance, x, is roughly 30 meters, we were able to get the error down to about one percent.

## Feb 2016 effort

a=306cm
b=232cm
c=258cm
After algebra: x = 3036 cm
Compare this with 137 ft using angular size.

## Footnotes and references

* ↑ Movie taken by 13hartc (discuss • contribs) 18:34, 11 February 2015 (UTC)
# Physics and Astronomy Labs/The Leyden jar and the Van de Graff generator
## Is this safe? No!

$C=\epsilon _{0}{\frac {\epsilon _{r}A}{d}}$
A is area, d is separation,
    $\epsilon _{0}=8.854\times 10^{-12}$ in SI units
For glass, the dilectric constant is $\epsilon _{r}\approx 10$ in SI units (actually between 3.8 and 14.5 according to Google)
The stored energy is
$U={\frac {1}{2}}CV^{2}\approx 2.15\,Joules$
where C is capacitance and V is voltage, estimated using the rule of thumb of 40kV/cm for air breakdown. In the above calculation, we took 80 kV, and a 3 millimeter gap.
The consensus is that it is 2.15 Joules. Student labs are not 100% reliable, but here are the numbers as they called them in:

### From the internet

It seems that 10 Joules is dangerous. For that reason I would never touch 2 Joules, especially if I wasn't sure that it was exactly 2 Joules! Here is what I read on two internet sites:
* "Warning: A capacitor or capacitor bank capable of discharging 25 J in less than 3 seconds, or 10 J in less than 0.5 seconds, can be lethal."
* Also found was this rambling answer:

## lab

Warning: This is a dangerous lab. Only a trained instructor should touch the equipment. A leyden jar can hold a dangerous charge even after you attempt to discharge it by making a spark.
Today we did a thrilling lab that was safe only because we took steps to keep it safe. A Leyden jar connected to a Van der Graff generator dramatically demonstrates the distintion between high voltage at low current and low voltage at high current.
* A Van der Graff generator by itself generates a spark of over 120kV (kilovolts), as can be verified by estimating the spark length and using the rule-of-thumb rule of 60kV/inch for breakdown of air.
* Connecting the Leyden jar reduces the spark length to about an inch, presumably because it takes so long to charge the extra capacitor that leakage inhibits the maximum voltage. But what a spark it makes!
There are safety concerns that need to be balanced before this becomes a Wikiversity resource. As we sport this out, we will post a video and images showing the spark, taken by user:13hartc.
The first of these images was recently drawn by our resident illustrator c:user:MarsLoveslions
* Dissectible_Leyden_jar.png
* Leyden_jar_showing_construction.png
* ↑ http://www.csgnetwork.com/dieconstantstable.html
* ↑ http://www.stephstuff.com/esafe/StoredEnergy.htm
# OpenStax University Physics

OpenStax | Quizbank | Physics equations | Quizbank attribution
This page is dedicated to the OpenStax textbook on the calculus-based Unviversity Physics. The algebra/trig bases textbook is OpenStax College Physics.

## Volume 2: Classroom-ready quizzes and exams are available on Wikiversity

* Quizbank/Electricity and Magnetism (calculus based): Short quizzes based on chapter examples in Volume 1 Unit 2.
* Quizbank/Electricity and Magnetism (calculus) Exams: Two 45 minute exams based on the aforementioned quizzes.
* Quizbank/University Physics Semester 2: Supplements the previous two exam banks to include optics for a one-semester course.

## Equation sheets and summaries are available for much of this book

Volume 1:
* OpenStax University Physics/V1
* OpenStax University Physics/V1/Ch 13 P 37: torque
* OpenStax University Physics/V1/Equations
* OpenStax University Physics/V1/Equations (master)
* OpenStax University Physics/V1/Formulas
* OpenStax University Physics/V1/Formulas (master)
Volume 2:
* OpenStax University Physics/V2
Electromagnetism:
* Quizbank/Electricity and Magnetism (calculus based)/Equations

## Accessing the textbook

* Download pdf files Volume_1 Volume_2 Volume_3
* Link to same pdf files Volume_1 Volume_2 Volume_3
These are probably not the latest version. The following online versions are probably newer:
* Volume 1:https://cnx.org/contents/1Q9uMg_a@11.1:Gofkr9Oy@17/Preface
* Volume 2: https://cnx.org/contents/eg-XcBxE@10.1:Gofkr9Oy@18/Preface
* Volume 3: https://cnx.org/contents/rydUIGBQ@10.1:Gofkr9Oy@19/Preface
While the latest version is typically the "best", some (but certainly not all) instructors may prefer to keep an older version until all changes have been incorporated into the course. Wikiversarians should feel free to update these links. Instructors who wish to "freeze" a certain edition need to place those links on a different Wikiversity page with instructions to not upgrade.February 2018 (UTC)

## Power point slides converted to PDF format for easy online use

These slides were provided by OER. The titles do not exactly match the chapters

### Volume 1

c:File:MUPHYS1 00 Physics 1 (Course Overview).pdf c:File:MUPHYS1 01 Units and Measurements (Module Overview).pdf c:File:MUPHYS1 02 Vectors (Module Overview).pdf c:File:MUPHYS1 03 Motion Along a Straight Line (Module Overview).pdf c:File:MUPHYS1 04 Motion in Two and Three Dimensions (Module Overview).pdf c:File:MUPHYS1 05 Newton s Laws of Motion (Module Overview).pdf c:File:MUPHYS1 06 Applications of Newton s Laws (Module Overview).pdf c:File:MUPHYS1 07 Work and Kinetic Energy (Module Overview).pdf c:File:MUPHYS1 08 Potential Energy and Convservation of Energy (Module Overview).pdf c:File:MUPHYS1 09 Linear Momentum and Collisions (Module Overview).pdf c:File:MUPHYS1 10 Fixed Axis Rotation (Module Overview).pdf 11-18 under construction.

#### Volume 2

c:File:MUPHYS2 00 Physics 2 (Course Overview).pdf

## Subpages

Special:Prefixindex/OpenStax University Physics/
# OpenStax University Physics/Archived sample/Syllabus
### T1

1/4 from Special:Permalink/1863415 to QB/c07energy_lineIntegral 2/4 from Special:Permalink/1863337 to QB/a18ElectricChargeField_findE 3/13 from Special:Permalink/1863397 to QB/c18ElectricChargeField_lineCharges 2/5 from Special:Permalink/1863338 to QB/a19ElectricPotentialField_Capacitance 2/4 from Special:Permalink/1863339 to QB/a19ElectricPotentialField_KE_PE 2/3 from Special:Permalink/1863399 to QB/c19ElectricPotentialField_SurfaceIntegral

### T2

3/6 from Special:Permalink/1863398 to QB/c19ElectricPotentialField_GaussLaw 2/4 from Special:Permalink/1863340 to QB/a20ElectricCurrentResistivityOhm_PowerDriftVel 5/21 from Special:Permalink/1863341 to QB/a21CircuitsBioInstDC_circAnalQuiz1 2/5 from Special:Permalink/1863342 to QB/a21CircuitsBioInstDC_circuits

### T3

1/13 from Special:Permalink/1863397 to QB/c18ElectricChargeField_lineCharges 1/21 from Special:Permalink/1863341 to QB/a21CircuitsBioInstDC_circAnalQuiz1 1/4 from Special:Permalink/1863343 to QB/a21CircuitsBioInstDC_RCdecaySimple 2/4 from Special:Permalink/1863344 to QB/a22Magnetism_forces 1/2 from Special:Permalink/1863345 to QB/a23InductionACcircuits_Q1 4/8 from Special:Permalink/1863346 to QB/a25GeometricOptics_image 2/4 from Special:Permalink/1863347 to QB/a25GeometricOptics_thinLenses

### T4

1/4 from Special:Permalink/1863337 to QB/a18ElectricChargeField_findE 1/4 from Special:Permalink/1863347 to QB/a25GeometricOptics_thinLenses 2/4 from Special:Permalink/1863348 to QB/a25GeometricOptics_vision 4/6 from Special:Permalink/1828922 to QB/c22Magnetism_ampereLaw 2/4 from Special:Permalink/1863400 to QB/c22Magnetism_ampereLawSymmetry 2/4 from Special:Permalink/1863401 to QB/c24ElectromagneticWaves_displacementCurrent

### FE

1/4 from Special:Permalink/1863337 to QB/a18ElectricChargeField_findE 1/13 from Special:Permalink/1863397 to QB/c18ElectricChargeField_lineCharges 1/6 from Special:Permalink/1863398 to QB/c19ElectricPotentialField_GaussLaw 1/5 from Special:Permalink/1863338 to QB/a19ElectricPotentialField_Capacitance 1/4 from Special:Permalink/1863339 to QB/a19ElectricPotentialField_KE_PE 1/3 from Special:Permalink/1863399 to QB/c19ElectricPotentialField_SurfaceIntegral 1/4 from Special:Permalink/1863340 to QB/a20ElectricCurrentResistivityOhm_PowerDriftVel 1/21 from Special:Permalink/1863341 to QB/a21CircuitsBioInstDC_circAnalQuiz1 1/5 from Special:Permalink/1863342 to QB/a21CircuitsBioInstDC_circuits 1/4 from Special:Permalink/1863344 to QB/a22Magnetism_forces 1/2 from Special:Permalink/1863345 to QB/a23InductionACcircuits_Q1 1/4 from Special:Permalink/1863347 to QB/a25GeometricOptics_thinLenses 1/6 from Special:Permalink/1828922 to QB/c22Magnetism_ampereLaw 1/4 from Special:Permalink/1863400 to QB/c22Magnetism_ampereLawSymmetry 1/4 from Special:Permalink/1863401 to QB/c24ElectromagneticWaves_displacementCurrent
# Literature/1995

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Subpages

* Literature/1995/Baxter $^{[^]}$
* Literature/1995/Card $^{[^]}$
* Literature/1995/Hirschheim $^{[^]}$
* Literature/1995/Kroto $^{[^]}$
* Literature/1995/Negroponte $^{[^]}$
* Literature/1995/Putnam $^{[^]}$
* Literature/1995/Reder $^{[^]}$
* Literature/1995/Smith $^{[^]}$
# Literature/1995/Yamanishi

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Authors
### Kenji Yamanishi

    NEC Research Institute, Inc., 4 Independence Way, Princeton, NJ

## Chronology

* Yamanishi, Kenji (1995). "Randomized Approximate Aggregating Strategies and Their Applications to Prediction and Discrimination," COLT '95: Proceedings of the Eighth Annual Conference on Computational Learning Theory, New York, NY: ACM. pp. 83-90. ISBN:0-89791-723-5 doi>10.1145/225298.225308 $^{[^]}$
* Literature/1975/Park $^{[^]}$
# Literature/1995/Cunningham

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Wikimedia

* w: Ward Cunningham
* w: WikiWikiWeb.
| w: wiki | w: Portland Pattern Repository |

## Chronology

* Reagle Jr., Joseph Michael (2010). Good Faith Collaboration: The Culture of Wikipedia. MIT Press. $^{[^]}$
* Literature/2001/Sanger $^{[^]}$
* Cunningham, Ward (1995). WikiWikiWeb. Portland Pattern Repository, Cunningham & Cunningham, Inc. http://c2.com/cgi/wiki $^{[^]}$
* Literature/1990/Berners-Lee $^{[^]}$
* Literature/1987/Atkinson $^{[^]}$
* Trigg, Randall Hagner & Mark Weiser (1986). "TEXTNET: A Network-based Approach to Text Handling," ACM Transactions on Information Systems (TOIS) Vol. 4, No.1 (Jan. 1986) pp. 1-23. $^{[^]}$
* Weyer, Stephen A. & Alan H. Borning (1985). "A Prototype Electronic Encyclopedia," ACM Transactions on Information Systems (TOIS) vol. 3, no. 1 (Jan. 1985) pp. 63-88. $^{[^]}$
* Literature/1984/Trigg $^{[^]}$
* Shneiderman, Ben (1983). "Direct Manipulation: A Step Beyond Programming Languages," Computer, Vol. 16, No. 8 (August 1983) pp. 57-69. $^{[^]}$
* Literature/1982/Brown $^{[^]}$
* Literature/1981/Kornfeld $^{[^]}$
* Literature/1980/Berners-Lee $^{[^]}$
* Soergel, Dagobert (1977). "An Automated Encyclopedia: A Solution of the Information Problem?" International Classification, 4(1): 4-10; 4(2): 81-89. $^{[^]}$
* Kochen, Manfred, ed. (1975). Information for Action: from Knowledge to Wisdom. New York: Academic Press. $^{[^]}$
* Kochen, Manfred (1972). "WISE: A World Information Synthesis and Encyclopaedia." Journal of Documentation, 28: 322-341. $^{[^]}$
# Literature/1996

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Subpages

* Literature/1996/Behe $^{[^]}$
* Literature/1996/Dale $^{[^]}$
* Campbell-Kelly, Martin & William Aspray (1996). Computer: A History of the Information Machine. Basic Books. $^{[^]}$
* Literature/1996/Gaines $^{[^]}$
* Literature/1996/Huntington $^{[^]}$
* Literature/1996/Newell $^{[^]}$
* Literature/1996/Oakhill $^{[^]}$
* Literature/1996/Wildman $^{[^]}$
# Literature/1996/Campbell-Kelly

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Excerpts

* During the second half of the 1980s, the joys of 'surfing the net,' began to excite the interest of people beyond the professional computer-using communities [...] However, the existing computer networks were largely in government, higher education and business. They were not a free good and were not open to hobbyists or private firms that did not have access to a host computer. To fill this gap, a number of firms such as CompuServe, Prodigy, GEnie, and America Online sprang up to provide low cost network access [...] While these networks gave access to Internet for e-mail (typically on a pay-per-message basis), they did not give the ordinary citizen access to the full range of the Internet, or to the glories of gopherspace or the World Wide Web. In a country whose Constitution enshrines freedom of information, most of its citizens were effectively locked out of the library of the future. The Internet was no longer a technical issue, but a political one. (p. 298)
* Lecture notes

## Wikimedia
### w: Computer: A History of the Information Machine

| w: World Wide Web | w: World Brain |

## Chronology

* Reagle Jr., Joseph Michael (2010). Good Faith Collaboration: The Culture of Wikipedia. MIT Press. $^{[^]}$
* Wallace, Danny P. (2007). Knowledge Management: Historical and Cross-Disciplinary Themes. Libraries Unlimited. $^{[^]}$
* Gillies, James & Robert Cailliau (2000). How the Web was Born: The Story of the World Wide Web. Oxford University Press. $^{[^]}$
* Literature/1999/Berners-Lee $^{[^]}$
* Campbell-Kelly, Martin & William Aspray (1996). Computer: A History of the Information Machine. Basic Books. $^{[^]}$
* Literature/1990/Berners-Lee $^{[^]}$
* Licklider, J. C. R. (1960). "Man-Computer Symbiosis." In: IRE Transactions on Human Factors in Electronics, vol. HFE-1, pp. 4–11. $^{[^]}$
* Bush, Vannevar (1945). "As We May Think." The Atlantic Monthly (July 1945): 101-108. $^{[^]}$
* Wells, H. G. (1938). World Brain. Garden City, NY: Doubleday, Doran & Co. $^{[^]}$
# Literature/1996/Pask

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Wikimedia
### w: Gordon Pask#Interactions of Actors Theory

* ... he remarked IA was meant to stand AI, Artificial Intelligence, on its head. Pask believed it was the job of cybernetics to compare and contrast. His IA theory showed how to do this. Heinz von Foerster called him a genius, "Mr. Cybernetics", the "cybernetician's cybernetician".
| w: Conversation Theory w: Interaction of actors w: Autopoiesis w: Humberto Maturana w: Francisco Varela w: Stafford Beer w: Nichlas Luhmann | w: Self-organization w: Norbert Wiener w: W. Ross Ashby w: Per Bak w: Stafford Beer w: David Bohm w: Manfred Eigen w: J. Doyne Farmer w: Heinz von Foerster w: Hermann Haken w: F. A. Hayek w: Francis Heylighen w: Steven Berlin Johnson w: Stuart Kauffman w: J. A. Scott Kelso w: Paul Krugman w: Nichlas Luhmann w: Heinz Pagels w: Gordon Pask w: Ilya Prigogine w: Peter Schuster |

## Chronology

* Pask, Gordon (1996). "Heinz von Foerster's Self-Organisation, the Progenitor of Conversation and Interaction Theories." Systems Research, 13(3): 349–362. $^{[^]}$
* Pask, Gordon (1993). Interactions of Actors (IA), Theory and Some Applications.
* Pask, Gordon (1976). Conversation Theory, Applications in Education and Epistemology. Elsevier.
* Pask, Gordon (1975). Conversation, Cognition and Learning. Elsevier. $^{[^]}$
* Pask, Gordon (1975). The Cybernetics of Human Learning and Performance. Hutchinson.
# Literature/1997

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Subpages

* Literature/1997/Arbib $^{[^]}$
* Literature/1997/Capra $^{[^]}$
* Literature/1997/Deutsch $^{[^]}$
* Literature/1997/Goodman $^{[^]}$
* Literature/1997/Johnson $^{[^]}$
* Literature/1997/Lesk $^{[^]}$
* Literature/1997/Ludlow $^{[^]}$
* Literature/1997/Rey $^{[^]}$
* Literature/1997/Stonier $^{[^]}$
# Literature/1996/Slaughter

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Excerpts

* The term 'critical' is often misunderstood [...]. However, it does not simply mean 'to criticize'. Nor does it signify a negative or derivative stance. It is not threatening and should not be construed as such. Rather, it signifies a range of methods and tools through which we may look 'beneath the surface' of social reality in order to realize the full potential of futures work. Critical futures study does recognize the partiality of traditions, cognitive frameworks and ways of knowing. It is therefore possibile to problematize aspects of the existing social and economic order and to explore some of their contradictions. (p. 139)

## Wikimedia
### w: Richard Slaughter

| w: Future studies w: World Brain | w: World Future Society w: World Futures Studies Federation |

## Chronology

* Barbour, Karen (2011). Dancing Across the Page: Narrative and Embodied Ways of Knowing. Intellect Books, 2011. $^{[^]}$
* Slaughter, Richard A., ed. (1996). New Thinking for a New Millennium (Futures and Education Series). Routledge, 1996. $^{[^]}$
* Kochen, Manfred, ed. (1989). The Small World: A Volume of Recent Research Advances Commemorating Ithiel de Sola Pool, Stanley Milgram, Theodore Newcomb. Norwood, NJ: Ablex Publishing Corp. (January 1, 1989). $^{[^]}$
* Kochen, Manfred, ed. (1975). Information for Action: from Knowledge to Wisdom. New York: Academic Press. $^{[^]}$
* Orwell, George (1949). Nineteen Eighty-Four. London: Secker and Warburg, 1949. $^{[^]}$
* Bernal, J. D. (1939). The Social Function of Science. London: Routledge & Kegan Paul Ltd. $^{[^]}$
* Wells, H. G. (1938). World Brain. Garden City, NY: Doubleday, Doran & Co. $^{[^]}$
* Wells, H. G. (1933). The Shape of Things to Come. Hutchinson. $^{[^]}$
* Huxley, Aldous (1932). Brave New World. Harper Perennial, 1932. $^{[^]}$
* Wells, H. G. (1932). "Wanted: Professors of Foresight!" Reprinted in Futures Research Quarterly. Vol. 3 No. 1 (Spring 1987): 89-91. $^{[^]}$
* Wells, H. G. (1923). Men Like Gods. Cassell and Co., Ltd. $^{[^]}$
* Wells, H. G. (1914). The World Set Free. Macmillan & Co. $^{[^]}$
* Wells, H. G. (1905). A Modern Utopia. New York, NY: Penguin Group, 2005. $^{[^]}$
# Riemann integral/x of 0 to 1/Explicit via staircase integrals/Exercise

Determine the definite integral
    $\int _{0}^{1}tdt$
explicitly with upper and lower staircase functions.
# Riesel conjectures
## Definition

For the original Riesel problem, it is finding and proving the smallest k such that k×b$^n$-1 is not prime for all integers n ≥ 1 and GCD(k-1, b-1)=1.

### Extended definiton

Finding and proving the smallest k such that (k×b$^n$-1)/GCD(k-1, b-1) is not prime for all integers n ≥ 1.
# Riemann integral/x^2 of 0 to 1/Explicit via step integrals/Example

We consider the function
    $f\colon [0,1]\longrightarrow \mathbb {R} ,t\longmapsto t^{2},$
which is strictly increasing in this interval. Hence, for a subinterval $[a,b]\subseteq [0,1]$, the value $f(a)$ is the minimum, and $f(b)$ is the maximum of the function on this subinterval. Let $n$ be a positive natural number. We partition the interval $[0,1]$ into the $n$ subintervals $\left[i{\frac {1}{n}},(i+1){\frac {1}{n}}\right]$, $i=0,\ldots ,n-1$, of length ${\frac {1}{n}}$. The step integral for the corresponding lower step function is
    $\sum _{i=0}^{n-1}{\frac {1}{n}}{\left(i{\frac {1}{n}}\right)}^{2}={\frac {1}{n^{3}}}\sum _{i=0}^{n-1}i^{2}={\frac {1}{n^{3}}}{\left({\frac {1}{3}}n^{3}-{\frac {1}{2}}n^{2}+{\frac {1}{6}}n\right)}={\frac {1}{3}}-{\frac {1}{2n}}+{\frac {1}{6n^{2}}}\,$
(see exercise for the formula for the sum of the squares). Since the sequences ${\left(1/2n\right)}_{n\in \mathbb {N} }$ and ${\left(1/6n^{2}\right)}_{n\in \mathbb {N} }$ converge to $0$, the limit for $n\rightarrow \infty$ of these step integrals equals ${\frac {1}{3}}$. The step integral for the corresponding step function from above is
    $\sum _{i=0}^{n-1}{\frac {1}{n}}{\left((i+1){\frac {1}{n}}\right)}^{2}={\frac {1}{n^{3}}}\sum _{i=0}^{n-1}(i+1)^{2}={\frac {1}{n^{3}}}\sum _{j=1}^{n}j^{2}={\frac {1}{n^{3}}}{\left({\frac {1}{3}}n^{3}+{\frac {1}{2}}n^{2}+{\frac {1}{6}}n\right)}={\frac {1}{3}}+{\frac {1}{2n}}+{\frac {1}{6n^{2}}}\,.$
The limit of this sequence is again ${\frac {1}{3}}$. By fact, the upper integral and the lower integral coincide, hence the function is Riemann-integrable, and for the definite integral we get
    $\int _{0}^{1}t^{2}\,dt={\frac {1}{3}}\,.$
# Riemann integral/x^3 of 1 to 2/Explicit via staircase integrals/Exercise

Determine the definite integral
    $\int _{1}^{2}t^{3}dt$
explicitly with upper and lower staircase functions.