How do you know the town in Bret Harte's "The Outcasts of Poker Flat" is hypocritical?

Readers know the residents of Poker Flat are hypocritical fairly early on in the story. We are told Mr. Oakhurst notices a moral shift in the town from the previous day. We are not told what kind of moral shift until a few lines later. Readers are then told the townspeople are now influenced by Biblical morals they normally ignore.  

There was a Sabbath lull in the air which, in a settlement unused to Sabbath influences, looked ominous.

What is very hypocritical of the townspeople of Poker Flat is they do not assume this high moral standing until they lose quite a bit of money and property to Oakhurst, Shipton, and the others. As long as those characters do not hurt Poker Flat, the townspeople are perfectly happy with the amoral standards of the visitors and the town itself.  

It had lately suffered the loss of several thousand dollars, two valuable horses, and a prominent citizen. It was experiencing a spasm of virtuous reaction, quite as lawless and ungovernable as any of the acts that had provoked it.

The hypocrisy continues as the townspeople pass judgment on the exiled group. The group may not return to Poker Flat. If they do, they risk their lives. Of course, by sending the group into the wilderness with minimal supplies, the townspeople are putting Oakhurst and the others at risk, too. For being people of such a "moral" town, Poker Flat's residents do not seem too worried about sending people to their likely death. That's hypocritical.

What were the major factors in the expansion of the middle class between 1920-1970?

There were several factors that helped the middle class grow between 1920-1970. One of these factors was the availability of good paying jobs. As factories expanded and the demand for products grew, there was an increased demand for workers in the manufacturing industries. Unions also increased their power during this time, and they were able to negotiate good pay, decent benefits packages, and improved working conditions. More professional jobs were available. This also helped boost pay and increase the number of people in the middle class.

As workers received more pay, they demanded more products. Some people were able to build or buy their own homes. The number of people that owned their homes increased during this time period. Other people were able to invest their money in businesses and industries. New inventions produced timesaving products that gave people more free time. Irons, refrigerators, and dishwashers were some examples. People wanted to get these inventions. Factors like these helped the economy grow even more.

With people making more money, demanding more products, and with a growing economy, our middle class was able to grow between 1920-1970.

In "The Pit and the Pendulum" by Edgar Allan Poe, why is the narrator afraid to open his eyes when he awakes to find himself on his back with no...

The narrator is afraid because he does not know where he is. 

"The Pit and the Pendulum" is a chilling horror story of a man who is the victim of the Spanish Inquisition.  He has been condemned to death, but he has no idea how he will be killed.  The terror of not knowing what is happening or going to happen consumes the narrator. 

Since his capture, the narrator keeps fainting and waking up.  When he wakes up, he is afraid to look around him.  He does not know what he will find, and he does not want to find out.  If he opens his eyes, everything will become more real. 

There I suffered it to remain for many minutes, while I strove to imagine where and what I could be.  I longed, yet dared not to employ my vision.  I dreaded the first glance at objects around me.  It was not that I feared to look upon things horrible, but that I grew aghast lest there should be nothing to see. 

When he does open his eyes, he does not feel any relief.  The room is so dark that, even with his eyes open, he gets no answers.  The narrator finds this frightening, too:

The blackness of eternal night encompassed me. I struggled for breath. The intensity of the darkness seemed to oppress and stifle me. The atmosphere was intolerably close. I still lay quietly, and made effort to exercise my reason.  

Eventually, the narrator realizes the pendulum is swinging toward him and will eventually cut him in half, so his situation will not really improve.  Sometimes knowing does not help.  Even though the narrator knows there is a pendulum, he does not know when it will finally kill him.  It is a terrible torture, actually.  He has to lay there for days, and he knows not how long, waiting for death.

Why does Harper Lee have Tom Robinson die? What does his death do for the plot?

In Harper Lee's To Kill a Mockingbird, Tom Robinson's death comes as a bit of a shock. Immediately after the trial, there is the (slim) possibility of getting an appeal, but then Atticus abruptly announces Tom Robinson was shot and killed while trying to escape prison. Though there are many valid opinions regarding why Lee had Tom Robinson die, I believe she included this plot point to illustrate the hopelessness and despair of the black community in racist Maycomb. Tom Robinson tried to escape because he knew he was never going to get a fair trial; his attempt was an example of extreme desperation. By including this moment in the plot, Lee illustrates the hopelessness a black individual feels in a white, racist community, and she further emphasizes the extent of the obstacles facing those trying to fight racial prejudice.

What is an example of parallelism in Of Mice and Men?

Parallelism in literature is the use of similar sentence or word structures to compare two things, ideas, or people. At the very beginning of Of Mice and Men, Steinbeck uses the following two parallel sentences, each beginning with the word "both," to compare Lennie and George: "Both were dressed in denim trousers and in denim coats with brass buttons. Both wore black, shapeless hats and both carried tight blanket rolls slung over their shoulders" (page numbers vary by edition). 

The author's repetition of the word "both" at the beginning of two consecutive sentences (and in later in the second sentence) sets up a parallel structure, and the purpose of this structure is to draw comparisons between Lennie and George. They are both dressed in denim and have similar hats and carry a blanket roll. In the sentences that follow, Steinbeck draws contrasts between George, who is "small and quick," and Lennie, who is "a huge man, shapeless of face." The author's use of parallel structures makes the reader immediately visualize the similarities between the two main characters before he delineates their differences. This type of structure also makes the reader feel like the characters are walking towards him or her; when the characters are farther away, the reader can only see their similarities in dress, but when the characters draw closer, the reader can see their distinctions. 

Why does Mr. Putnam hold a grudge against Francis Nurse?

Mr. Putnam holds a grudge against Francis Nurse because Nurse had taken part in a land war with his neighbors, and one of those neighbors was a Putnam.  The conflict grew to battle-like proportions, lasting for two days in the middle of the woods.  Nurse had originally rented his land, but he worked hard to eventually be able to purchase it, and many people resented his rise in status.  Further, years ago, Mr. Putnam's wife's brother-in-law, a man named Bayley, had been up for a Salem ministry position, and though he had all the qualifications and a two-thirds vote, he was prevented from taking office by a faction that included the now-powerful Nurse family.  Moreover, some other families whose land was adjacent to the Nurse homestead joined with them to create Topsfield, "a new and independent entity whose existence was resented by Old Salemites" like the Putnams.

`int_(pi/4)^(pi/2) cot^3(x) dx` Evaluate the integral

`int_(pi/4)^(pi/2)cot^3(x)`

Let's evaluate the indefinite integral by rewriting the integrand as,

`intcot^3(x)=intcot(x)cot^2(x)dx`

Now use the identity:`cot^2(x)=csc^2(x)-1`

`=intcot(x)(csc^2(x)-1)dx`

`=int(cot(x)csc^2(x)-cot(x))dx`

`=intcot(x)csc^2(x)dx-intcot(x)dx`

Now let's evaluate `intcot(x)csc^2(x)dx` by integral substitution,

Let `u=cot(x)`

`=>du=-csc^2(x)dx`

`intcot(x)csc^2(x)dx=intu(-du)`

`=-intudu`

`=-u^2/2`

substitute back `u=cot(x)`

`=-1/2cot^2(x)`

Use the common integral `intcot(x)dx=ln|sin(x)|`

`:.intcot^3(x)dx=-1/2cot^2(x)-ln|sin(x)|+C` , C is a constant

Now let' evaluate the definite integral,

`int_(pi/4)^(pi/2)cot^3(x)dx=[-1/2cot^2(x)-ln|sin(x)|}_(pi/4)^(pi/2)`

`=[-1/2cot^2(pi/2)-ln|sin(pi/2)|]-[-1/2cot^2(pi/4)-ln|sin(pi/4)|]`

`=[-1/2*0-ln(1)]-[-1/2(1)^2-ln(1/sqrt(2))]`

`=[0]+1/2+ln(1/sqrt(2))`

`=1/2+ln(2^(-1/2))`

`=1/2-1/2ln(2)`

`=1/2(1-ln(2))`