The Math Trick That College Will Never Teach You! : Parametric Integration

The math trick that college will never teach you! : Parametric Integration

Parametric integration is one such technique that once you are made aware of it, you will never for the love of god forget it. It goes by many names : ‘Differentiation under the Integral sign’, ‘Feynman’s famous trick’ , ‘Parametric Integration’ and so on.

Let me demonstrate :

Now this integral might seem familiar to you if you have taken a calculus course before and to evaluate it is rather simple as well.

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Knowing this you can do lots of crazy stuff. Lets differentiate this expression wrt to the parameter in the integral – s (Hence the name parametric integration ). i.e 

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Look at that, by simple differentiation we have obtained the expression for another integral. How cool is that! It gets even better.

Lets differentiate it once more: 

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.

.

.

If you keep on differentiating the expression n times, one gets this : 

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Now substituting the value of s to be 1, we obtain the following integral expression for the factorial. This is known as the gamma function.

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There are lots of ways to derive the above expression for the gamma function, but parametric integration is in my opinion the most subtle way to arrive at it. :D

This is a really powerful technique and I strong suggest that if you have taken calculus, then please do read this article.

Have a great day!

EDIT:  It had to be gamma(n+1) not gamma(n) .Thank you @mattchelldavis

More Posts from Philosophical-amoeba and Others

7 years ago

Tough as a Tardigrade

Without water, a human can only survive for about 100 hours. But there’s a creature so resilient that it can go without it for decades. This one millimeter animal can survive both the hottest and coldest environments on Earth, and can even withstand high levels of radiation. This is the tardigrade, and it’s one of the toughest creatures on Earth, even if it does look more like a chubby, eight-legged gummy bear. 

Tough As A Tardigrade

Most organisms need water to survive. Water allows metabolism to occur, which is the process that drives all the biochemical reactions that take place in cells. But creatures like the tardigrade, also known as the water bear, get around this restriction with a process called anhydrobiosis, from the Greek meaning life without water. And however extraordinary, tardigrades aren’t alone. Bacteria, single-celled organisms called archaea, plants, and even other animals can all survive drying up.

Tough As A Tardigrade

For many tardigrades, this requires that they go through something called a tun state. They curl up into a ball, pulling their head and eight legs inside their body and wait until water returns. It’s thought that as water becomes scarce and tardigrades enter their tun state, they start synthesize special molecules, which fill the tardigrade’s cells to replace lost water by forming a matrix. 

Tough As A Tardigrade

Components of the cells that are sensitive to dryness, like DNA, proteins, and membranes, get trapped in this matrix. It’s thought that this keeps these molecules locked in position to stop them from unfolding, breaking apart, or fusing together. Once the organism is rehydrated, the matrix dissolves, leaving behind undamaged, functional cells.

Tough As A Tardigrade

Beyond dryness, tardigrades can also tolerate other extreme stresses: being frozen, heated up past the boiling point of water, high levels of radiation, and even the vacuum of outer space. This has led to some erroneous speculation that tardigrades are extraterrestrial beings.

Tough As A Tardigrade

While that’s fun to think about, scientific evidence places their origin firmly on Earth where they’ve evolved over time. In fact, this earthly evolution has given rise to over 1100 known species of tardigrades and there are probably many others yet to be discovered. And because tardigrades are so hardy, they exist just about everywhere. They live on every continent, including Antarctica. And they’re in diverse biomes including deserts, ice sheets, the sea fresh water, rainforests, and the highest mountain peaks. But you can find tardigrades in the most ordinary places, too, like moss or lichen found in yards, parks, and forests. All you need to find them is a little patience and a microscope.

Tough As A Tardigrade

Scientists are now to trying to find out whether tardigrades use the tun state, their anti-drying technique, to survive other stresses. If we can understand how they, and other creatures, stabilize their sensitive biological molecules, perhaps we could apply this knowledge to help us stabilize vaccines, or to develop stress-tolerant crops that can cope with Earth’s changing climate. 

Tough As A Tardigrade

And by studying how tardigrades survive prolonged exposure to the vacuum of outer space, scientists can generate clues about the environmental limits of life and how to safeguard astronauts. In the process, tardigrades could even help us answer a critical question: could life survive on planets much less hospitable than our own?

From the TED-Ed Lesson Meet the tardigrade, the toughest animal on Earth - Thomas Boothby

Animation by Boniato Studio


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8 years ago
Harmonograph, H. Irwine Whitty, 1893
Harmonograph, H. Irwine Whitty, 1893
Harmonograph, H. Irwine Whitty, 1893
Harmonograph, H. Irwine Whitty, 1893
Harmonograph, H. Irwine Whitty, 1893
Harmonograph, H. Irwine Whitty, 1893
Harmonograph, H. Irwine Whitty, 1893

Harmonograph, H. Irwine Whitty, 1893

“The facts that musical notes are due to regular air-pulses, and that the pitch of the note depends on the frequency with which these pulses succeed each other, are too well known to require any extended notice. But although these phenomena and their laws have been known for a very long time, Chladni, late in the last century, was the first who discovered that there was a connection between sound and form.”

source here


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7 years ago
Useless Indonesian Chiken Facts: Ayam Cemani

Useless Indonesian Chiken facts: Ayam Cemani

Hi tumblr, meet one of the most expensive chiken In the world from Indonesia; Ayam Cemani

No we not paint it black with our excessive coal resources

It simply have unique genes that makes it whole body, meat and even organs all black

It’s chiken so yeah you can eat it, and it said to be very delicous. thou all these years living in Indonesia, I never actually tried it to tell you how it taste

I mean.. It cost up to Rp. 100 Million or about 10.000 USD for one of this chiken…so…..

The extravagant price mostly come from  myths that surround these beautiful chiken. said it’ll bring you luck, health and every benefits you can imagine people would said about this emo chiken

pic sc: err…Internet

- J -


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7 years ago

The Pirate Queen of the South China Sea

The Red Flag Fleet under Ching Shih’s rule went undefeated, despite attempts by Qing dynasty officials, the Portuguese navy, and the East India Company to vanquish it. After three years of notoriety on the high seas, Ching Shih finally retired in 1810 by accepting an offer of amnesty from the Chinese government. Ching Shih died in 1844, at the ripe old age of 69.At the dawn of the 1800s, a former prostitute from a floating brothel in the city of Canton was wed to Cheng I, a fearsome pirate who operated in the South China Sea in the Qing dynasty. Though the name under which we now know her, Ching Shih, simply means “Cheng’s widow,” the legacy she left behind far exceeded that of her husband’s. Following his death, she succeeded him and commanded over 1,800 pirate ships, and an estimated 80,000 men.

Her husband, Ching I, was the formidable commander of the Red Flag Fleet of pirate ships. He married a 26-year-old Ching Shih in 1801. She quickly took to the pirate life and when Ching I died six years later, Ching Shih wasn’t going to let Ching I’s adopted son, Cheung Po Tsai take over.  Cheung Po Tsai, however, was more than just Ching Shih’s adopted son –-  the young man had also been Ching I’s lover.

Keep reading


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7 years ago

A whole lot of books

A Whole Lot Of Books

In 1610, after successfully rebuilding, reopening and firmly establishing the Bodleian Library, Sir Thomas Bodley made it his new goal to ensure the library’s long-term relevance. He forged an agreement with the Stationers’ Company that saw his library receive one copy of everything printed under royal license. This would guarantee that the Bodleian would continue to be at the cutting edge of literature and librarianship.

And this was when the Bodleian became, by effect, the first Legal Deposit library in the UK. From 1662, the notions were solidified and the Royal Library and the University Library of Cambridge were also party to the same privileges. Until the establishment of the British Library in 1753, the university libraries of Oxford and Cambridge - especially the Bodleian Library, which had enjoyed the legal deposit privilege the longest – served as the de facto national libraries of the United Kingdom.

But what’s the current state of Legal Deposit at the Bodleian? And, as @LowlandsLady asked us on Twitter, might it ever feel like a curse to have this multitude of books rolling into the library?

There’s nobody better to answer these questions than Jackie Raw, Head of Legal Deposit Operations in our Collections Management department. From this point on, this Tumblr post belongs to Jackie.

The Legal Deposit Libraries Act 2003 obliges publishers to deposit, at their own cost, one copy of every printed publication that is published or distributed in the UK with the British Library and, upon request, with up to five other Legal Deposit Libraries of which the Bodleian is one.

These Legal Deposit Libraries seek to ensure that the UK published output is preserved for posterity. To do this the requesting libraries together use the Agency for the Legal Deposit Libraries. The Agency makes claims for titles on our behalf and, on receipt, these are recorded and distributed weekly to the libraries. By collaborating the Legal Deposit Libraries seek to develop and co-ordinate national coverage and preservation of, and access to, publications acquired by legal deposit.

A Whole Lot Of Books

A random selection of books arriving by Legal Deposit.

While the Legal Deposit system has existed in various forms since 1662, the reformed Act in 2003 extended the provision for printed material to cover non-print works. This brings new and emerging publishing media under its scope. The Regulations for this came into force in April 2013. We now have the opportunity to add e-books, e-journals, digital maps and digital music scores to our collections and to harvest and archive content from UK-published material on UK websites under the new legislation.

Again the libraries are working together to ensure coverage. The benefits of holding a collection of over 12 million physical items, much of which has been deposited under Legal Deposit, and a growing digital collection cannot be overstated.

A Whole Lot Of Books

A recent count of books arriving daily at the Bodleian Libraries.

Legal Deposit ensures the nation’s heritage is collected systematically and preserved for posterity. It supports and advances the teaching and research activities of the University of Oxford and national and international scholarship more widely by collecting, recording and making available this material and it provides, in a cost-effective way, access to a wide range of publications many of which are unlikely to be found outside of the Legal Deposit system.

However, it is also a responsibility.

Clearly there can be pressures on space, storage and transportation. There are staff levels and processing costs to consider and issues around funding limitations. On the other hand there is provision within the Act to allow libraries, other than the British Library, to be selective in what we acquire. So, by consulting and collaborating we aim to ensure as complete coverage as possible within resourcing limits without overburdening any one library with the rate at which our collections are growing.


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8 years ago

Aphasia: The disorder that makes you lose your words

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It’s hard to imagine being unable to turn thoughts into words. But, if the delicate web of language networks in your brain became disrupted by stroke, illness or trauma, you could find yourself truly at a loss for words. This disorder, called “aphasia,” can impair all aspects of communication. Approximately 1 million people in the U.S. alone suffer from aphasia, with an estimated 80,000 new cases per year.  About one-third of stroke survivors suffer from aphasia, making it more prevalent than Parkinson’s disease or multiple sclerosis, yet less widely known.

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There are several types of aphasia, grouped into two categories: fluent (or “receptive”) aphasia and non-fluent (or “expressive”) aphasia. 

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People with fluent aphasia may have normal vocal inflection, but use words that lack meaning. They have difficulty comprehending the speech of others and are frequently unable to recognize their own speech errors. 

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People with non-fluent aphasia, on the other hand, may have good comprehension, but will experience long hesitations between words and make grammatical errors. We all have that “tip-of-the-tongue” feeling from time to time when we can’t think of a word. But having aphasia can make it hard to name simple everyday objects.  Even reading and writing can be difficult and frustrating.

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It’s important to remember that aphasia does not signify a loss in intelligence. People who have aphasia know what they want to say, but can’t always get their words to come out correctly. They may unintentionally use substitutions, called “paraphasias” – switching related words, like saying dog for cat, or words that sound similar, such as house for horse. Sometimes their words may even be unrecognizable.  

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So, how does this language-loss happen? The human brain has two hemispheres. In most people, the left hemisphere governs language.  We know this because in 1861, the physician Paul Broca studied a patient who lost the ability to use all but a single word: “tan.” During a postmortem study of that patient’s brain, Broca discovered a large lesion in the left hemisphere, now known as “Broca’s area.” Scientists today believe that Broca’s area is responsible in part for naming objects and coordinating the muscles involved in speech. Behind Broca’s area is Wernicke’s area, near the auditory cortex. That’s where the brain attaches meaning to speech sounds. Damage to Wernicke’s area impairs the brain’s ability to comprehend language. Aphasia is caused by injury to one or both of these specialized language areas.

Fortunately, there are other areas of the brain which support these language centers and can assist with communication.  Even brain areas that control movement are connected to language. Our other hemisphere contributes to language too, enhancing the rhythm and intonation of our speech. These non-language areas sometimes assist people with aphasia when communication is difficult.

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However, when aphasia is acquired from a stroke or brain trauma, language improvement may be achieved through speech therapy.  Our brain’s ability to repair itself, known as “brain plasticity,” permits areas surrounding a brain lesion to take over some functions during the recovery process. Scientists have been conducting experiments using new forms of technology, which they believe may encourage brain plasticity in people with aphasia.  

Meanwhile, many people with aphasia remain isolated, afraid that others won’t understand them or give them extra time to speak. By offering them the time and flexibility to communicate in whatever way they can, you can help open the door to language again, moving beyond the limitations of aphasia.


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7 years ago
“Men Want To Beat Me. I Play Men, Six, Seven Hours A Day. Men… They Do Not Beat Me.”
“Men Want To Beat Me. I Play Men, Six, Seven Hours A Day. Men… They Do Not Beat Me.”
“Men Want To Beat Me. I Play Men, Six, Seven Hours A Day. Men… They Do Not Beat Me.”
“Men Want To Beat Me. I Play Men, Six, Seven Hours A Day. Men… They Do Not Beat Me.”
“Men Want To Beat Me. I Play Men, Six, Seven Hours A Day. Men… They Do Not Beat Me.”
“Men Want To Beat Me. I Play Men, Six, Seven Hours A Day. Men… They Do Not Beat Me.”
“Men Want To Beat Me. I Play Men, Six, Seven Hours A Day. Men… They Do Not Beat Me.”

“Men want to beat me. I play men, six, seven hours a day. Men… they do not beat me.”

- Masako “Katsy” Katsura, first woman to compete for a world’s title in billiards

The “First Lady of Billiards” learned from her brother-in-law and in the 50s became Japan’s only female billiards pro. She paved the way for women to come, then left the spotlight and lived a quiet life. She died in 1995.


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9 years ago
This Is What Happens When A Carrot Is Fired At 300 Km/hour At An Egg, Through Two Sheets Of Cardboard. 

This is what happens when a carrot is fired at 300 km/hour at an egg, through two sheets of cardboard. 

This is what happens if you separate out the two sheets:

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The egg survives! This shows how a Whipple shield works, and is what spacecraft use to protect themselves from micrometeoroid impacts in space. When the projectile (in this case a carrot, but in space it could be a speck of paint, a piece of an old satellite, or a bit of space rock) hits the first layer, it’s moving so fast that it starts to vaporise, because the energy of the collision is enough to break almost every bond in the substance.

It then sprays outwards, spreading the force of impact across a much wider area, meaning the second layer can stop it going any further, keeping your egg (or astronauts) safe.

Watch the full video on our YouTube channel.


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9 years ago
Image: The Libertine Falstaff Sits With A Woman On His Lap And A Tankard In His Hand In An Illustrated

Image: The libertine Falstaff sits with a woman on his lap and a tankard in his hand in an illustrated scene from one of William Shakespeare’s Henry IV plays. (Kean Collection/Getty Images)

The eggplant and peach emoji are standard code for racy thoughts these days, but people have been using food as sexual innuendo for centuries. In fact, William Shakespeare was a pro at the gastronomic double entendre [insert blushing face emoji here]. The Salt blog asked a Shakespeare expert to help them decode some of the bard’s bawdy food jokes. (And the result is delightful. – Nicole)

50 Shades Of Shakespeare: How The Bard Used Food As Racy Code


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philosophical-amoeba - Lost in Space...
Lost in Space...

A reblog of nerdy and quirky stuff that pique my interest.

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