`y = x, y = (2x)/(1 + x^3)` (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve...

The shell has the radius `x - (-1)` , the cricumference is `2pi*(x+1)` and the height is `(2x)/(1+x^3) - x,` hence, the volume can be evaluated, using the method of cylindrical shells, such that:

`V = 2pi*int_(x_1)^(x_2) (x+1)*((2x)/(1+x^3) - x) dx`

You need to find the endpoints, using the equation `(2x)/(1+x^3) - x = 0 => 2x - x - x^4 = 0 => 3x - x^4 = 0 => x(3 - x^3) = 0...

The shell has the radius `x - (-1)` , the cricumference is `2pi*(x+1)` and the height is `(2x)/(1+x^3) - x,` hence, the volume can be evaluated, using the method of cylindrical shells, such that:

`V = 2pi*int_(x_1)^(x_2) (x+1)*((2x)/(1+x^3) - x) dx`

You need to find the endpoints, using the equation `(2x)/(1+x^3) - x = 0 => 2x - x - x^4 = 0 => 3x - x^4 = 0 => x(3 - x^3) = 0 => x = 0 ` and `x = root(3) 3`

`V = 2pi*int_0^(root(3) 3) (x+1)*((3x - x^4)/(1+x^3)) dx`

`V = 2pi*int_0^(root(3) 3) (x+1)*(3x - x^4)/((x+1)(x^2-x+1)) dx`

Reducing the like terms yields:

`V = 2pi*int_0^(root(3) 3) (3x - x^4)/(x^2-x+1) dx`

`V = 2pi*(int_0^(root(3) 3) (3x)/(x^2-x+1) -int_0^(root(3) 3) (x^4)/(x^2-x+1) dx`

`V = 2pi*(int_0^(root(3) 3) (3x)/(x^2-x+1)dx - int_0^(root(3) 3) x^2 dx - int_0^(root(3) 3) x dx + int_0^(root(3) 3) (x)/(x^2-x+1)dx)`

`V = 2pi*(int_0^(root(3) 3) (4x)/(x^2-x+1)dx - int_0^(root(3) 3) x^2 dx - int_0^(root(3) 3) x dx)`

`V = 2pi*(2*int_0^(root(3) 3) (2x+1-1)/(x^2-x+1)dx - x^3/3|_0^(root(3) 3) - x^2/2|_0^(root(3) 3))`

`V = 2pi*(2*int_0^(root(3) 3) (2x-1)/(x^2-x+1)dx+ 2*int_0^(root(3) 3) (1)/(x^2-x+1)dx - 1 - (root(3) 9)/2)`

You need to solve `int_0^(root(3) 3) (2x-1)/(x^2-x+1)dx ` using substitution `x^2-x+1 = t => (2x-1)dx =dt.`

`V = 2pi*(2*ln(x^2-x+1)|_0^(root(3) 3)+ 2*int_0^(root(3) 3) (1)/((x-1/2)^2 + ((sqrt3)/2)^2)dx - 1 - (root(3) 9)/2)`

`V = 2pi*(2*ln((root(3) 9)-(root(3) 3)+1)+ (4/sqrt3)*arctan (2x-1)/sqrt3|_0^(root(3) 3) - 1 - (root(3) 9)/2)`

`V = 2pi*(2*ln((root(3) 9)-(root(3) 3)+1)+ (4/sqrt3)*arctan (2(root(3) 3)-1)/sqrt3 -(4/sqrt3)*(pi/6) - 1 - (root(3) 9)/2)`

Hence, evaluating the volume, using the method of cylindrical shells, yields `V = 2pi*(2*ln((root(3) 9)-(root(3) 3)+1)+ (4/sqrt3)*arctan (2(root(3) 3)-1)/sqrt3 -(4/sqrt3)*(pi/6) - 1 - (root(3) 9)/2)`

Grim says "poor kevin" and gives a "long sorrowful look". How do you think grim's name and these two actions foreshadow what's to come in the story?

You are smart to recognize Grim’s words and actions as foreshadowing. Foreshadowing, of course, is a mention in a novel of what could happen later. When Grim talks about “poor Kevin” and gives a “long sorrowful look,” it foreshadows Kevin’s ultimate death from his birth defects. (Further, Gram also calls Kevin “poor baby.” This continues the foreshadowing.) Max recognizes that there is something wrong with Kevin. Kevin was always the kid at daycare who wore...

You are smart to recognize Grim’s words and actions as foreshadowing. Foreshadowing, of course, is a mention in a novel of what could happen later. When Grim talks about “poor Kevin” and gives a “long sorrowful look,” it foreshadows Kevin’s ultimate death from his birth defects. (Further, Gram also calls Kevin “poor baby.” This continues the foreshadowing.) Max recognizes that there is something wrong with Kevin. Kevin was always the kid at daycare who wore “shiny braces on his crooked legs,” but who also referred to himself as Robot Man. In this instance, however, Max emphatically defends his friend who is both adventurous and imaginative. Max is proud of Kevin, especially when they team up as the dual character of “Freak the Mighty.” Fortunately (or unfortunately), Grim and Gram can see the harsh reality: Kevin is a severely crippled child who will probably not live very long. Sure enough, Kevin’s heart is “too big for his body,” and Kevin eventually dies. Max remains so proud of Kevin that Max ends up writing all of their adventures down so that we can read about them.

Find the amount accumulated in the increasing annuities. $110 deposited monthly for 6 years at 4 % per year. ( Assume the end of period deposits...

Since $110 are deposited monthly, then $110*12 = $1320 are deposited at the end of each year, for the next 6 years in an account paying 4% per year compounded annualy.

To find the value accumulated in increasing annuities you need to take a look at each of the $1320 payment.

Hence, `P = $1320, n = 5, i = 4/100` and the formula used is `A =P(1+i)^n`

The first payment will produce a compound...

Since $110 are deposited monthly, then $110*12 = $1320 are deposited at the end of each year, for the next 6 years in an account paying 4% per year compounded annualy.

To find the value accumulated in increasing annuities you need to take a look at each of the $1320 payment.

Hence, `P = $1320, n = 5, i = 4/100` and the formula used is `A =P(1+i)^n`

The first payment will produce a compound amount of

`1320(1 + 4/100)^5 = 1320*1.04^5`

You need to use n=5 instead n=6 since the money is deposited at the end of the first year and earns interest for only 5 years.

Hence, the future value of annuity is:

`1320*1.04^5 + 1320*1.04^4 + 1320*1.04^3 + 1320*1.04^2 + 1320*1.04^1 + 1320`

Notice that the terms of the sum are the terms of a geometric sequence, having the ratio q = 1.04 and the first term b = 1320.

`S = b*(q^n-1)/(q-1)`

`S = 1320*(1.04^6 -1)/(1.04 - 1)`

`S = 1320*(0.265319018496)/(0.04)`

`S ~~ $8755.527`

Hence, evaluating the amount accumulated in the increasing annuities yields `S ~~ $8755.527.`

What effects does starting the story with a quatrain in Latin have on your impression of the story?

The Latin quatrain, without translation, gives the story a feeling of mystery, religious overtones, and gravitas. For someone who doesn't know Latin, if the translation is not provided, reading the quatrain results in confusion and uncertainty. This is not unlike the narrator's state in the first paragraph, for he seems to be in a partial dream state where he does not fully comprehend or recognize the voices around him. Second, if one recognizes the words...

The Latin quatrain, without translation, gives the story a feeling of mystery, religious overtones, and gravitas. For someone who doesn't know Latin, if the translation is not provided, reading the quatrain results in confusion and uncertainty. This is not unlike the narrator's state in the first paragraph, for he seems to be in a partial dream state where he does not fully comprehend or recognize the voices around him. Second, if one recognizes the words as Latin, one may have the feeling of being in a Latin mass in a Catholic church. Again, this religious atmosphere matches the setting of the first paragraph because the man is on trial by the Spanish Inquisition, an institution of the Catholic Church. Finally, Latin generally impresses a reader, signifying a rich history and depth of learning. Latin is a very old language, so the quatrain gives the story a feeling of history and solemnity, both consistent with the mood Poe wants to create in the story. Even if the reader does not know Latin at all, the quatrain helps establish the mystery, religious context, and historical feeling of the story that Poe was aiming for.

What kind of person is Ozymandias as he is depicted in Percy Bysshe Shelley's poem of the same name?

We learn something about Ozymandias from line three of the poem. These lines provide a description of the individual whose image has been sculpted in stone, which now lies broken in the sand.

Near them, on the sand,Half sunk, a shattered visage lies, whose frown,And wrinkled lip, and sneer of cold command,

The words in bold inform us that the sculpture expresses a frown which suggests a serious expression; 'wrinkled lip' informs of a...

We learn something about Ozymandias from line three of the poem. These lines provide a description of the individual whose image has been sculpted in stone, which now lies broken in the sand.

Near them, on the sand,
Half sunk, a shattered visage lies, whose frown,
And wrinkled lip, and sneer of cold command,

The words in bold inform us that the sculpture expresses a frown which suggests a serious expression; 'wrinkled lip' informs of a haughty expression, possessed by one who regards others with contempt. This is further supported and accentuated by the word 'sneer', which tells us that the person so depicted had disdain for those whom he commanded. The fact that his command is described as 'cold' suggests that he was heartless and cruel. Our perception is therefore of a cruel, hard, ruthless taskmaster who led without any love for his subjects. We can therefore rightly assume that he must have been either a dictator or tyrant.

The speaker tells us that the sculptor 'well those passions read,' which is an indication that the skilled artist was not remiss in the manner in which he portrayed his subject in this now decayed work. The line "The hand that mocked them, and the heart that fed" further informs us that Ozymandias saw his subjects as buffoons and treated them as if they were idiots. He relished abusing his subjects and he fed his overblown ego by treating them with utter disregard and making fools of them.

Further insight is provided into Ozymandias' unpleasant superciliousness in the lines:

And on the pedestal these words appear:
‘My name is Ozymandias, king of kings:
Look on my works, ye Mighty, and despair!'

So vain and egotistical was he that he expressed his greatness on the pedestal of his statue, stating that he was greater than any ruler. Even the mightiest of the mighty could not challenge his glory for he was so all-powerful and great that all any other ruler could do was to become disparaged when they witnessed his magnitude and magnificence.

It is therefore ironic that all that has remained of Ozymandias' so-called prodigious power is a broken statue, enveloped by the sands of the desert.

 Nothing beside remains

Ozymandias has been defeated by death and time. The lonely, open and vast desert has become his final resting place, leaving a poor testament to his once, as he believed, incomparable might.

What are some related poems, short stories, or graphics related to "The Bet?"

This is a challenging question. It is hard to think of any literary works that are directly related to or derived from "The Bet." But there are numerous works that have some resemblance to one or more of the themes in Chekhov's story. Those themes are solitude, study, and murder. 

In the first place, there are the four gospels in the New Testament: Matthew, Mark, Luke and John. The lawyer read many books during his period of captivity, but eventually he seems to have focused his full attention on the Christian message contained in those four books.

Then after the tenth year, the prisoner sat immovably at the table and read nothing but the Gospel. It seemed strange to the banker that a man who in four years had mastered six hundred learned volumes should waste nearly a year over one thin book easy of comprehension. 

The implication is that after the prisoner had looked for what he wanted to find out in six hundred volumes, he found it in the Bible. The best translation of the four gospels is still the King James Version because of its beautiful language.

Robinson Crusoe by Daniel Defoe is related to "The Bet" in that Crusoe was in a sort of solitary confinement on a deserted island for many years and became a philosopher as a result of the experience.

"The Prisoner of Chillon," a beautiful poem by Lord Byron, is about a man who was kept a prisoner in a dungeon for many years. During a large part of his captivity he was alone because the relatives he had been imprisoned with died off and left him in effect in solitary confinement. He too became a philosopher in his confinement.

The Count of Monte Cristo by Alexandre Dumas and Auguste Maquet is about a man who spends a great many years in solitary confinement in the Chateau D'If on an island in the Mediterranean.

Poems about solitude include Edgar Allan Poe's "Alone" and William Butler Yeats' "The Lake Isle of Innisfree." Yeats said that his poem was inspired by his reading of Henry David Thoreau's book Walden. Thoreau loved solitude and read a great many profound books himself. Like the lawyer in Chekhov's story "The Bet," Thoreau was trying to solve the mystery of human existence. In the chapter titled "Solitude" in Walden he wrote:

To be in company, even with the best, is soon wearisome and dissipating. I love to be alone. I never found the companion that was so companionable as solitude. 

In a little-known short story by Jack London titled "The Star Rover," a prisoner in solitary confinement who undergoes frequent torture over the years learns to use his mind as an escape from his misery. He experiences past lives and other fascinating psychological phenomena.

In Leo Tolstoy's short story "God Sees the Truth But Waits," a man is wrongfully imprisoned for murder and spends many years in prison but develops religious enlightenment which gives him wisdom, patience and solace.

There must be many other literary works about the themes of solitude and religious enlightenment. A story that concerns a man like Chekhov's banker who plans to commit a murder to get out of paying a legitimate debt is the Sherlock Holmes classic "The Adventure of the Speckled Band." There are other Sherlock Holmes stories in which the villain's motive is to get out of parting with money.

What effect did the large number of berries have on Brian in Hatchet?

Brian gets a stomach ache from eating too many berries.

One of Brian’s first concerns when he finds himself the only survivor of a plane crash in the Canadian wilderness is where to find food.  Brian has only a hatchet and some clothing.  He does not have any food.  When daydreaming about his next meal, he decides to look for berries.

The sun was still high and that meant that he had some time before...

Brian gets a stomach ache from eating too many berries.

One of Brian’s first concerns when he finds himself the only survivor of a plane crash in the Canadian wilderness is where to find food.  Brian has only a hatchet and some clothing.  He does not have any food.  When daydreaming about his next meal, he decides to look for berries.

The sun was still high and that meant that he had some time before darkness to find berries. He didn't want to be away from his—he almost thought of it as home—shelter when it came to be dark. (Ch. 6)

Brian does find berries, but they are not very tasty.  They are tart and make his mouth dry.  He eats them anyway because he is hungry and there is nothing else.  He hasn’t eaten in two days.  The berries turn his stomach, but he eats too many of them just the same.

Brian wakes up with serious stomach pains.  The berries, which he later names gut cherries, are almost not worth the price.

It was as if all the berries, all the pits had exploded in the center of him, ripped arid tore at him. He crawled out the doorway and was sick in the sand, then crawled still farther and was sick again, vomiting and with terrible diarrhea for over an hour, for over a year he thought, until he was at last empty and drained of all strength. (Ch. 7)

Although the plan to look for berries and use the birds to locate them is a good one, the berries that Brian finds are hardly helpful.  He realizes that he has to eat them, but he tries to focus on the ripe ones and only eat them a few at a time.  Brian finds some raspberries, which are much better, but for them he has competition from a bear.