Integral Calculus

Math may seem to some people as being a jumbled mess of illogical expressions and rules, it can be scary to some, confusing to many, boring to others, some consider it to be pure torture and the ones that are able to focus will discover its elegant insights and these are the people that will find it fun to solve mathematical problems.  Math is a language and calculus can be considered to be a dialect of math.  Calculus topics are a bit like grammar, many people are able to communicate clearly without worrying about all the details, but the details do exist, and they are in place for good reasons.  Using proper grammar is very important in communication, because it is an aspect of the language, and without proper grammar you will never really be sure whether or not you understand what someone is saying or what you are reading.

Ordinary Calculus includes simple methods of doing differential calculus and integral calculus.  I made up the name Ordinary Calculus just to distinguish it from more Advanced Calculus.  I think that the biggest problem with calculus is that teachers and books start using calculus before the students actually understand what calculus actually is.  The main problem with calculus is that many given concepts are associated with other concepts, and these may be defined in terms of the original concept.  This running around in circles, ends up building and expanding the original idea, without actually providing an understanding or giving an insight on the concept behind the idea.  Concepts that are defined with circular associations, are not much help for students because they don’t actually reveal anything, and this type of teaching does not allow students to develop a real good understanding of calculus.  A circular definition uses the term that is being defined as a part of the definition, and this is done because it assumes the reader or student has a prior understanding of this term.  Students who are taught this way may be able to solve problems, but many students that have taken calculus actually never comprehended the deep meaning which is hidden beneath the surface under layer after layer of circular definitions.

I am going to assume that you know what a function is and what a derivative is, and if you do, then you can learn about the integral.  While differential calculus focuses on rates of change, such as slopes of tangent lines and velocities, integral calculus deals with total size or value, such as lengths, areas, and volumes.  Integral calculus helps in finding the antiderivatives of a function.  These antiderivatives are also called the integrals of the function.  The process of finding the antiderivative of a function is called integration.  The inverse process of finding derivatives is finding the integrals.  The integral of a function represents a family of curves.  An integral is basically the exact opposite of a derivative.  An integral gives the area under a curve from the x-axis to the curve from a specified range.  An integral is expressed by the integration symbol.  The integration process is used to obtain the function f(x) from knowledge of its derivative.  This is shown pictorially below.

When you want to determine how something is changing, then you will work with the derivative and when you want to see how much something has changed, then you deal with the integral.  This symbol looks kind of like a tall skinny S, and this ∫ is called an integral.  The integral sign is associated with “summa”, or sum.  An integral is an area or the generalization of area.  Integration is the process of finding a function, when you have been given its derivative.  When you use either differentiation or integration, you don’t get a specific answer, you will only get another equation from which you may obtain the specific answer you are looking for.  Integrals are the values of the function found by the process of integration.  The process of getting f(x) from f'(x) is called integration.  Integrals assign numbers to functions so that they can be used to describe displacement and motion problems, area and volume problems, and so on that arise by combining all the small data.

Written for FOWC with Fandango – Integral.

Kiss Me with Those Fuzzy Lips

Uncle Steve went to the zoo with his brother Joe and his family.  It was a lovely day, but Joe’s wife Kathie never liked Steve being around her children, 6-year-old Christine and 4-year-old Billy, because he was an incorrigible character that would forever be a bachelor, because no good woman would ever have him.  Kathie had a look of disgust on her face when they went to visit the giraffes and Steve said, “Kiss me with those fuzzy lips”, so she ushered the kids away and said, “Let’s go visit the gorillas, that should be fun.”  Steve laughed and told his brother, “Doesn’t she look stunning and you know how much I have always preferred a full-grown bush over a trimmed pussy.”   Joe told his brother, “I know all of your stories about your mustache rides and I just wish that you could behave better around the kids.”  Steve seemed to be confused and he asked Joe, “What did I say that was so wrong, it is not like I was talking about a Dirty Sanchez?  We should get a couple of beers before we catch up to your family.”

Written for FOWC with Fandango – Dirty and for Stine Writing – Poetry, Positivity, and Connecting!, Simply 6 Minutes hosted by Christine Bialczak.

Sliding Into Someone’s Dms

This is my realm, as I enjoy direct messaging a stranger or acquaintance, by using a social networking platform.  My intention is to garner a response and I could be reaching out for a romantic connection, a friendship, or just as a way to connect with someone, but not to be creepy.  I’m no stranger to the DM slide and I have had some success using it, but I have also had a few failures.  Since many people choose to share their Instagram handles on their dating app profiles, a simple search and some creativity on your end, can land you in their inbox.

I always avoid mentioning where I found them and no one has ever asked.  Instead, I just use my opening line and place it in the posts on their Instagram.  I never ask them a question that puts pressure on them to respond, as I like to keep it easy as opposed to demanding.  If they are interested, they will respond and keep the interaction going.  This is the line that I normally use and it has opened up a lot of doors for me.  “Hello, how are you doing?  My name is Jim and I find dating messages to be awkward, but if I don’t break the ice and contact anyone, then I won’t meet anyone.  It is hard to come up with something worthwhile to say when you don’t know a person yet.  Hit me back if you find me to be interesting.”  If they don’t write back, I didn’t waste that much time and I can DM direct message somebody else, so nothing ventured, nothing gained, but if my slide works out, then I have gained access to a new realm.  What starts out as a DM slide, has the possibility to turn into a really fun weekend fling.

Written for FOWC with Fandango – Realm.

Not in the Clique

The other kittens don’t want to accept me.  They spend all of their time together and they will not welcome me into the group.  I realize that there is some type of pecking order here which I have not figured out yet and I have no idea which one of my new housemates is going to become my friends or my competitors.  I am usually cautious of anything that is new and I will try to stay a healthy distance away from anyone with whom I have not had a positive experience with yet, but the other day when I arrived, they all hissed at me.  Maybe they think that I am going to eat up all of their treats, or they won’t like me using their litter box, or they are just jealous because I am so damn cute.

I am trying to have patience but the rest of these cats need to learn how to accept change.  There is no way of telling how things are going to work out between all of us, but I am willing to give this place another week before I am out of here.  I don’t mind being a stray, as there are plenty of lizards around here for me to eat.  I don’t care either way on whether or not they let me into their clique, as these cats have serious social problems and it is not my job to fix them.  I don’t want to be the target of their aggression anymore and I would be perfectly willing to stay here if you could just control the more belligerent cats, as this place looks like it could make a nice home.  I am OK with sharing this space with the other cats, but all of this hissing and growling needs to end.  We need to find some type of balance, especially while I am trying my best to acclimate to this new environment.

Written for FOWC with Fandango – Clique and for Sadje at Keep It Alive What Do You See #85.

Cicadas

These insects don’t do much besides providing nutrition for other animals and because they are so defenseless to their predators, they are considered manna from heaven.  Their sheer numbers help them to survive as a species, as eventually everything gets tired of eating them.  Thus, no matter how many cicadas get eaten, there will still be millions left to reproduce.  Some cicadas appear each summer, and then there are the ‘periodical cicadas’ which emerge on a regular schedule every 13 or 17 years, depending on the species.  Male cicadas make a hypnotic buzzing sound, which is their love song used to attract females for mating.  Cicadas can be annoying to some people, but this is one of the most amazing phenomena that occurs on our planet and it doesn’t last long, so people should enjoy it while they can.

Periodical cicadas spend almost their whole lives being only a foot or two underground, living on sap from tree roots.  When the mature cicada nymphs emerge, they mate and then die, and their newly hatched offspring drop to the ground and burrow in for the next 13 or 17 years.  The periodical cicadas emerge nearly simultaneously in great densities numbered in the trillions to spend a few weeks of life in the open air where they molt, buzzing males call for a mate to copulate with and after they have mated diligent females will deposit their fertilized eggs in nests.  The periodical cicadas begin life above ground as nymphs once they hatch.  These rice-grain sized nymphs immediately fall to the ground, where they burrow their way into what will be their home for the foreseeable future.  The years that follow are spent feeding on the juices of plant roots and slowly maturing in preparation for the upcoming emergence.

Evidence suggests that they don’t feel pain the way other creatures do, so if you are brave enough to try and eat these critters, you can celebrate a victory for being able to do something that makes others cringe, while you are enjoying a feast on these bugs.  There are plenty of cicada recipes that you can try like dry roasted or air fried, but I think the best way is to prepare them to be crispy and crunchy, but you may have to eat them with your eyes closed, otherwise you may have trouble swallowing them.

Groups of cicadas that share the same emergence years are known as broods.  This spring, bugs belonging to one of the biggest broods of 17-year cicadas, called Brood X or the Great Eastern Brood, are making their appearance.  So, love is in the air this spring in the eastern United States, where billions of cicadas are appearing in the biggest emergence event since 2004.  The Brood X cicadas will stretch from Tennessee to New York.  Bob Dylan was given an honorary degree by Princeton University in 1970, which was a year that the Brood X cicadas emerged.  He was moved by the experience to write the song ‘Day of the Locusts’, which was included on his eleventh studio album New Morning.

Written for FOWC with Fandango – Diligent, for June Monthly prompts – Crispy and crunchy and for Word of the Day – Victory.

The Rat’s Ass

I like to understand things, so I search stuff to try and figure out meanings.  In ancient Greece people were asked to sacrifice rat’s ass to the gods.  If someone didn’t have a rat’s ass to sacrifice, they were accused of not caring about the gods.  So, when someone says that they don’t give a rat’s ass, that means that they don’t care.  No, that story is bullshit as the phrase came about because of the way that it seems to roll off the tongue.  In the same manner, people say that they don’t give a flying fuck because of its alliterative appeal.  When you tell somebody that you don’t give two shits about them, you are leaving them with hope that you could possibly have one shit left for them.  If you do give two shits about someone that would mean that you care a little about them, but because nobody likes shit and it just confuses people.

Written for FOWC with Fandango – Search.

Adventure at Sea

Ruth, Eli and Chaman were traveling by ship on the Marah from Byblos to Rome and Chaman told them that two of the sailors on this ship were old friends of her husband Kamran, who was not making the trip with them.  Chaman told her new friends Ruth and Eli that her husband was the leader of a band of thieves that included Barabbas, Farhad and Omid, but Kamran got out of the business when he saw that he could make more money legally than working as a criminal.  Ruth, Eli and Chaman took a walk around the deck, when they ran into Farhad and Chaman said, “I have told my new friends about your unique ability to convey a sequence of events, bring your characters to live and explain all the vivid details, that transport your listeners deeply into your stories.”  Farhad said, “Oh, so you have told them that I like to spin yarns and that I am full of shit.”  Chaman said, “Well that is yet to be determined, but I always enjoy your stories and Eli and Ruth both said that they would also love to hear some of them.”

Farhad said, “Barabbas, Omid and I were all drinking in Mehragan’s saloon, when we met these Cilician pirates who tried to convince us into joining their group.  Barabbas said that he had no interest as he suddenly left with a woman that he met in the tavern.  Omid and I told them that we did not have any experience being aboard ships, but we were both interested in traveling to different places.  They seemed to be impressed with our pillaging skills and they said that they would be able to teach us all the necessary skills that we needed to be on a ship at sea, as long as we were prepared to attack anyone of any nationality, owing loyalty to no one, but our shipmates and the captain of the ship.  We both decided to take a chance and the next day when we woke up we were aboard the pirate ship Neptune’s Trident, captained by Skull Thrush of Tarsus.”

“We both enjoyed being out at sea, breathing the fresh air and listening to the tranquility of the water.  I liked being able to look out into sea and have dreams about visiting exotic lands.  Omid was adept at climbing, so he worked with the ship’s sails rigging and masts and I was assigned to the galley, because I had some experience cooking.  One day we attacked a ship that was carrying wine and olive oil from the Sardinian port of Caralis to the Phoenician port of Tyre.  We let all of our captives go after we confiscated their cargo and then we made our way toward Crete, because we knew that they would not ask any questions about where we got the cargo from.”

“We had a favorable breeze for several days, and everybody was looking forward to how we would spend our share of the booty after we sold the stolen cargo.  We broke into a cask of wine and started to enjoy our spoils.  Just after midnight our ship was attacked by a great sea monster.  Sea monsters are the most dangerous water-dwelling creatures that a crew can ever face, they are ultimately horrifying and mysterious, because these foes can come up on you all of a sudden and you never know when they are lurking just below the surface.  The deep seas and the oceans are able to hide many secret inhabitants and just because you have not seen something, you cannot say that it does not exist.  The unknown should always be feared, especially when it is enormous in size and resides in the vast dark unfathomable depths.  I woke up when I heard the cry, ‘Sea monster attacking off starboard bow!’  I got to the deck when the huge Sea Serpent was still a whole ship’s length distance away.”

“It was a full moon that night, which allowed all of us to see the monster approaching.  This sea monster was so huge that its head reached as high as our mast, when it came out of the water.  The colossal head and neck of this monster was only visible once it surfaced, but we discerned a wavy motion in the water behind the creature, so we knew that part of its body was concealed under water moving in undulations, like a snake or a sea serpent.  The enormous monster had a long-pointed snout, its teeth were glistening in the moonlight and its red glowing eyes were about the size of a warrior’s shield.  I saw the sea monster dive under water, and it raised its tail aloft, as it plunged back into the water.  I saw that its body was as wide as our ship, and about three times as long.  It came up again and spouted water from its snout like a whale, just as the creature moved slowly toward our ship.  I was never so terrified in my whole life, as I sensed the creature was going to attack us.”

“The creature thrashed its head and neck from side to side, as it climbed across the bow and then it bit off the head of one of our crew.  It chewed for a short time and then spit the skull back out on to the deck.  My fear was enhanced when I saw how easily this creature was able to kill us.  The sea monster continued its attack, as the pirates tried to stab it with their swords.  It swallowed up several more members of the crew and it did extensive damage to the ship as it crashed its massive body into the Neptune’s Trident.”

“I yelled out to Captain Skull Thrush, ‘Let’s head back to port, before the monster sinks us, as half the crew is already lost.’  Captain Skull Thrush replied, ‘Shut the hell up!  I can’t let that sea monster win!’  I noticed a split in the seam of the ship near the wood planks, which made up the hull.  Water started gushing in and we were sinking fast, as crew members were being devoured all around me.  The battle continued until captain Skull Thrush struck his sword into the animal’s right eye and the monster returned to the sea.  The sea serpent got away, diving below the water, leaving behind a large funnel that our ship was drawn into.  Omid and I were both thrown out of the whirlpool and I could hear the suction of the water along with a loud splintering noise of cracking wood.  This funnel caused our ship to swirl around very fast, which resulted in it breaking up into pieces.”

“I grabbed onto a plank of cedar, which kept me afloat and I paddled my way over towards Omid.  We were able to fasten together a tiny raft to lift ourselves out of the freezing water.  We saw four of our fellow pirates clinging on to another raft and the captain was among them.  Casks of wine and olive oil were floating all around us and then suddenly the monster resurfaced and looked straight at captain Skull Thrush.  The captain raised his sword and he was swinging it wildly, as the sea monster swallowed him in one gulp.  The three other brave men that were on the same raft as the captain, did their best to survive, but the monster ate every one of them.  As clouds rolled in front of the moon, it became very dark, but it was apparent that Omid and I were the only ones that were able to escape.  The sea was very rough and we were drifting aimlessly in the dark.”

“We managed to fall asleep and when we woke up the next morning, we were both hungry and very thirsty.  We were able to salvage a cask of wine from the sea that was floating near us.  We were stuck on the open sea without food and with only a slim to none chance of surviving.  I realized that our survival was dependent on us being able to find an island.  I felt that I had no other recourse, except to pray to God for His help, even though I knew that Omid was not a believer.  Omid said that he would not pray with me, but he hoped that all of my prayers would be answered.”

“I prayed, ‘Oh, God above we are stranded in your vast dark sea.  We ask for your help, power and wisdom and that you allow us to find the strength, so we can make it to safety.  May your wind carry us to a place where we can be rescued?  If we are allowed to survive by your good grace, then we will give up our thieving ways and work in peace with the universe.’  After drifting with sea currents for two days, we saw the broad tail of a fish appearing above the waves and it seemed to be following us.”

“A being surfaced next to our raft that had the torso of a woman, but below her navel, she was a fish.  I shook Omid and told him to look at the mermaid.  She had long green hair, the color of seaweed that was adorned with seashells.  Her skin was blue and pale and she wore a string of sparkling pearls that enhanced her breasts.  She smiled at us, but her eyes kept on blinking, as I think she was trying to adjust her sight to being above the water.  She was breathing heavily, hacking and coughing, because she was not used to being in the fresh air.  She dove back underwater and then she came up to the surface again.  She made a honking sound and then she seemed to be breathing better.  She spoke saying, ‘I am Analisa, daughter of the third cave of the Crab Nebula, the Holder of the Wondrous Comb and the Heir of the Radiant Pearls of the Boiling Sea’.”

“I told Analisa that my name was Farhad and then I introduced my good friend Omid.  I said that we were thieves from Byblos, who had just become pirates aboard the ship Neptune’s Trident.  I let Analisa know that I had just prayed to God and then I asked if He had sent her to assist us.  Analisa said, ‘What makes you think that God is a He?’  Then she said that she saw us fighting the sea monster, which was her enemy, so she wanted to help us.  She said that this sea monster had eaten many of her friends and also a lot of fish in this part of the sea.  She told us that this monster was a glutton, who ate so much that it was creating a famine.”

“Analisa said that she was grateful that we had chased the sea serpent away.  She said that she followed the monster to its lair and she could tell that it was severely injured, it was moaning in agony and it was probably mortally wounded, which made her very happy.  She brought us fish to eat and asked if we needed any help getting back to land.  I looked at the freshly caught fish and my mouth started watering, as I was real hungry, but I was still hesitant, as I had never eaten a raw fish before.  I saw Omid take out his knife and slice off the head of the fish and then he cut off a thin slice, which he immediately consumed.  I yelled at Omid to cut off a slice for me, which he did and then I ate it.  I told him to pass me the wine as I needed something to wash this down with.”

“Analisa asked us if she could have some of our wine.  I replied, ‘Certainly’ and then she grabbed the cask of wine and took a big gulp and then she sang, ‘Joy to all the fishes in the deep blue sea, God will provide answers if we let it be.  I like to frolic under the sea, its better there and I let down my hair.  I am happiest when wet and the fish are all around, emitting a soothing sound.  When I am floating under the sea, my life is filled with bubbles and I have no troubles.  I enjoy being on the ocean floor and I would never dream of going ashore.  I can play all day in the waves by the coral caves, far below and far away.  There is no air when you are way down there and I can live my life without a care.  A life lived without wine is very sad, and it may make you become raving mad.  The taste is sublime, it is almost divine, yes this is some mighty fine wine’.”

“I stood up on the raft and told Analisa what a beautiful voice she has and I let her know how much I enjoyed hearing her song.  Analisa smiled at me and then she swam over to me and kissed me on my lips, while she was holding her arms around me.  Omid said, ‘Don’t let her pull you under!’  Analisa broke off our kiss and she swam away from me, and she folded her tail underneath her, then she said, ‘Your friend does not trust me.  I understand that mermaids are known to lure handsome young sailors to fall in love with them and then they deliberately drag these sailors down into the water and drown them, as they squeeze the life out of them.  I came here to help you because you defeated my greatest foe.’  Analisa started combing her hair and then she said, ‘You can either accept my kindness or ask me to leave, but don’t accuse me of trying to harm you’.”

“I told Omid to apologize to Analisa, because we needed her help and she had been good to us.  Omid said he was sorry for his accusations and Analisa said that she would accept his apology and it was time to move on and get us to land.  She pushed our raft, till we finally landed on the island of Kasos.  We both thanked her, but she said that she owed us more than we owed her and then we waved goodbye to each other.  It was quite an ordeal and we were both feeling lucky to be alive, when we finally made it shore.  We drank from a stream on the island and we found figs, grapes, berries and birds to eat.  We were both alive and we had a fantastic tale of survival to tell.  Captain Pelles rescued us and we have been aboard the Marah ever since.”

Written for FOWC with Fandango – Mermaid.

Remove That Smile From Your Face!

Joe said, “Stop smiling, as that smirk of yours is irritating and you don’t have any good reason to be so pleased with yourself.  Nobody likes a smiker!  It is so freaking hot, this intense sun is scorching my skin and I am sweating like a pig.”  Eddie said, “What in the hell is a smiker?  Joe said, “Dude, did you not study English in High School, as a smiker is one who smirks.  When you go around smirking, people think that you are laughing at them, or taking some type of sarcastic pleasure at someone else’s unhappiness or misfortune.”

Eddie started singing, “Smirk, smirk, smirk, they call me the smiker.  If I want to be happy, I don’t need your permission and I have an awful lot of really good reasons to be happy.  I have two loving parents and my dog loves me also.  I have a date tonight with Cindy and we are going to see that new Jason Statham film Wrath of Man.  I was not smirking, as that is just the way my smile looks and if you want to know why I am smiling, it is because I was just thinking about ice cream and then that ice cream truck started coming up the street.  What do you say, how about me buying each of us an ice cream so we can cool off, OK bro?”  Joe said, “Sure buddy, as that sounds much better than splitting wood and honking geese.”  Eddie said, “Pass that joint over here, as you have already had too much.”

Written for FOWC with Fandango – Remove, for May Monthly Writing Prompts – Splitting wood and honking geese, and for Word of the Day Challenge Prompt – Smiker.

The Eureka Moment

There is a legend about Archimedes being asked to uncover a fraud that was being perpetrated against the Greek tyrant King Hieron II of Syracuse.  The king suspected that a goldsmith was cheating him when he ordered a solid gold crown, thinking that it was partly made of silver, so the local tyrant contracted the ancient Greek polymath Archimedes to detect any fraud in the manufacture of this golden crown.  Archimedes accepted the challenge, but he didn’t have any idea how he was going to resolve this, till he came up with Archimedes’ Principle.

Archimedes’ Principle says that when a body is partially or completely immersed in a liquid, the buoyant force acting on it is equal to the weight of the displaced liquid and it acts in the upward direction at the center of gravity of the displaced liquid.  This is the basis of why things float on water and such weight measurements can be used to determine the average density of irregular objects, and also to estimate body fat in people.  There is a story which says that Archimedes discovered the law of buoyancy while he was taking a bath, realizing that the water that ran over equaled in volume the submerged part of his body.  As he leaped out of the tub more water splashed on the floor, and he was so excited that he didn’t even stop to grab a towel, dashing out of his house running through the streets naked to announce his discovery, but in reality, he did further experiments.

Water has a specific gravity of 1, and the densities of all other materials are relative to that of water.  Gold has a higher density than silver, which results in gold objects feeling like they are heavier than silver.  Archimedes took two pieces of pure gold and two of pure silver that had identical weights to the weight of the king’s crown.  He then successively immersed the gold, the silver, and the crown in a container filled to the brim with water and measured the volume of water that overflowed with each material.  He found that the crown displaced more water than the gold but less than the silver, thereby proving that the crown contained some other metal which was less dense than gold.  The goldsmith had to make the crown larger when he substituted silver for some of the gold, so that it would weigh the same as a pure gold crown.  However, the silver-gold crown took up more space in the bowl and made the water rise higher.

Written FOWC with Fandango – Buoyant.

Not A Sure Bet

Twenty horses will line up in the starting gate for the Kentucky Derby today and the overall favorite is the number 14 horse Essential Quality, who is going off at 2:1 odds.  He won his last five races and he is certainly the one to beat.  There are some other contenders, but most of the horses running today are long shots, but hey that is where you can pick up some money, as the greater the odds, the better the payoff, but of course you have to win to get paid.  The number 15 horse Rock Your World has odds of 5-1, and the number 1 horse Known Agenda has odds of 6-1, number 9 Hot Rod Charlie is going off at 8-1 and for number 17 Highly Motivated the odds are set at 10-1.  A Sure Bet would allow you to benefit from the differences found between odds set by the bookmakers, but the only sure bet is not to make a bet.

Written FOWC with Fandango – Sure.