When do gases and liquids contract, and why do they contract?

The Kinetic Theory of Matter states that an object is made up of tiny particles - atoms or molecules - that are in constant motion. A book is well-defined - it has a certain volume and a certain mass. The book is made up of molecules of various chemicals making up the paper and the ink. While the book is not vibrating - not moving at all, in fact - the molecules inside it are in constant motion - they are vibrating in place. The same is true for a liquid. Water sitting in a glass might not appear to be moving, but the molecules in it are actually moving around and vibrating - to a much greater extent than solids. Molecules in gases are in an even greater constant motion. They have higher kinetic energies, and can actually take up any available space. Molecules/atoms in gases interact negligibly making this possible, as opposed to the free-moving molecules in liquids that are still interacting and hence giving it a defined volume. In solids, the molecules are very tightly packed that only the vibrations are possible.
However, even in solids, increasing the temperature will cause a bit of an expansion. This is the reason why there are tiny gaps in rail roads, for instance. It allows for the thermal expansion of the material. This happens because as temperature rises, the kinetic energy in the molecules also increases, and they vibrate more rapidly - this results in a displacement due to vibration that's higher than lower temperatures. The same happens for gases and liquids.
The reverse of this is also true. As temperature decreases, kinetic energy decreases. This would result to less movement. In solids, while there is still compression, it is not very noticeable as the molecules/atoms are already tightly packed. However, in liquids and gases, this is very significant. Lower temperature means lower kinetic energy and hence less vibration and the molecules can interact more. This results to contraction in liquids and gases. This is more dramatic in gases. According to the kinetic molecular theory of gases, the particles in a gas do not interact at all. This is because of high kinetic energy. However, by reducing that, this gives the particles a chance to interact with each other, thus further slowing them down, and you get contraction in lower temperatures - very much dramatic when you put a balloon in a freezer. This is also when it may start to condense.
Hence, in brief, liquids and gases contract or get compressed at lower temperatures. This is because of the lowering of the kinetic energy that lessens the vibrations and motions in the particles.
(Note: In some cases, like water, compression of a liquid stops at a certain point. When it freezes, ice is less dense than water. This is because of the structure of the water in the ice crystal).
http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch4/kinetic4.html

https://www.businessinsider.com/why-train-tracks-buckle-in-extreme-heat-2013-7

https://www.sdbor.edu/educators/praxis/Documents/bhsu/documents/ms_sc3.pdf

Single Variable Calculus, Chapter 5, 5.3, Section 5.3, Problem 18

Find the derivative of the function $\displaystyle y = \int^0_{\frac{1}{x^2}} \sin^3 t dt$ using the 1st Fundamental Theorem of Calculus.

Using Properties of Integral

$\displaystyle \int^a_b f(x) dx = - \int^b_a f(x) dx$

So we have

$\displaystyle \int^0_{\frac{1}{x^2}} \sin^3 t dt = - \int^{\frac{1}{x^2}}_0 \sin ^ 3 t dt$

Let $\displaystyle u = \frac{1}{x^2}, \frac{du}{dx} = - \frac{2}{x^3}$. Then,


$
\begin{equation}
\begin{aligned}

\frac{d}{dx} \int^{\frac{1}{x^2}}_0 \sin^3 t dt =& \frac{d}{dx} \left(\int^u_0 - \sin^3 t dt \right)
\\
\\
y' =& \frac{d}{du} \left(\int^u_0 - \sin ^3 t dt \right) \frac{du}{dx}
\\
\\
y' =& (- \sin ^3 u) \frac{du}{dx}
\\
\\
y' =& \left(- \sin ^3 \frac{1}{x^2}\right) \left( \frac{-2}{x^3} \right)
\\
\\
y' =& \frac{2}{x^3} \sin ^2 \left( \frac{1}{x^2} \right)

\end{aligned}
\end{equation}
$

What line of poetry should I use for my descriptive writing assignment?

What you need to find is a poem that paints only part of the painting, so to speak. Then you can feel free to take the prompt further on your own and describe the scene that you imagine, based on the few beginning words that the poet supplies. I have three recommendations. The text of each poem is provided in a link below.
If you pick Robert Frost’s “The Road Not Taken,” you can follow the first line, “Two roads diverged in a yellow wood.” How do you see these two roads? Are they at all the same? How are they different? Which one would you be inclined to take, if you had to make the choice? What lies at the end of each road?
If you pick William Wordsworth’s “I Wandered Lonely as a Cloud,” you can describe the field of daffodils that he discovers. Take the lines from the second verse, “They stretched in never-ending line / Along the margin of a bay.” How do you see the daffodils in relationship to the water? Are boats sailing nearby? Are any other people walking nearby? Does anyone else see these 10,000 daffodils besides you? Or is this a special place that only you know about? How would it look in summer, fall, or winter, without the flowers?
If you pick Robert Browning’s “My Last Duchess,” you can describe the painted portrait of the woman. Begin with the first two lines, “That’s my last Duchess painted on the wall, / Looking as if she were alive.” Since she was a Duchess, she was part of royalty. What country is she from? What is she wearing in the portrait? What color is her hair? Is she smiling or frowning? Is she holding anything important? Where is this painting hung? Who sees it on a regular basis?
Ask yourself questions, and let your imagination answer them. Then write down what you see in your mind.
https://www.poetryfoundation.org/poems/43768/my-last-duchess

https://www.poetryfoundation.org/poems/44272/the-road-not-taken

https://www.poetryfoundation.org/poems/45521/i-wandered-lonely-as-a-cloud

How does Beckett use repetition in Waiting for Godot? Why does it matter?

It is worth noting that the repetitive parallelism of the play's two acts has at least some variation in it. For instance, in the second act, the tree has sprouted some leaves, and Pozzo has regressed from a pompous master into a blind man dependent on Lucky. Thus, while a variety of repetition occurs throughout the play, there are small changes throughout. This quality contributes to a "same, but not quite" atmosphere, as Beckett crafts a decidedly weird space that is both always the same and subtly different. It is important to recognize this aspect of the play's repetitiveness because it contributes to the persistently meaningless or absurd tone. Despite the small differences we observe between the two acts, the story remains entirely inconsequential, and the characters' actions, though varying slightly, remain as futile as ever. As noted in the other answers to this question, the second act culminates in the same ending: Godot fails to show up. Thus, by offering repetitive events that vary slightly from one another but still add up to the same ending, Beckett enhances the absurd futility of his vision. Potential changes to the second act's trajectory result in the same frustrating stagnation as in act one.

Time is circular in Samuel Beckett's play Waiting for Godot, and the repetition of dialogue and events within the play serves to make this non-linear structure clear. Every single day, Vladimir and Estragon wait for Godot to arrive, going through the same motions and even repeating some of the exact same dialogue verbatim. Their lives are almost Sisyphean in their absurdity, waiting for someone who will never arrive. Beckett masterfully employs repetition to underscore the futility of Vladimir and Estragon's task (in as much as the act of waiting can be called a task) and the ridiculousness of their continued hope. I have seen productions of the play in which the repetition is almost ritualistic, as though Vladimir and Estragon are trying to impose some structure on their existences, and others in which it is played as though the two simply keep forgetting what they've said and done before. Ultimately, the repetition, like the fact that Godot never arrives and never will, is a commentary on the human condition.

As in a number of his other plays, Beckett attempts to sketch and pace a play which emulates the ebb and flow of life itself. Waiting for Godot is rife with false starts, false hopes, uncertainty, boredom, and the deep tenderness of friendship. The use of repetition plays a role in this allegorical depiction of everyday life and the human experience.
Early in the play, we see the following pieces of dialogue twice, absolutely identical, nearly back-to-back:

VLADIMIR: It hurts?
ESTRAGON: (angrily). Hurts! He wants to know if it hurts!

The text is the same, yet a line later the roles are reversed. It is Estragon who says "It hurts?" Why does Beckett subject his audience to this nonsensical repetition between Vladimir and Estragon? Didi and Gogo's relationship serves as a sounding board for a wide range of human interactions and feelings: they are married, they are brothers, they are rivals and friends. This repetitive dialogue reflects the sometimes redundant nature of close relationships, such as marriages or close friendships. You switch roles; in one moment one is indignant at the other, and then it changes, but somehow the relationship remains at an even keel. It's an observation on human relationships, namely that we tend to fall into the same little arguments time and again.
There is also, of course, the repetition of each day in Vladimir and Estragon's world: waiting for Godot. Each day they wait, and each day Godot does not arrive. Godot may represent many things: God, death, some kind of transformative change which may never arrive. This repetitive waiting mirrors the audience's everyday life. Our days often look like one another until some kind of large change impacts us, and then we must adapt. This incessant waiting, the daily questions of "What am I waiting for, and why?" exist in life just as they do in the play.

How does Johnson use repetition to support his meaning in the poem?

Jonson uses repetition and parallelism throughout his poem to enforce his meaning: namely, that the object of his affections pays too much mind to her physical appearance, which has the effect of obscuring her natural beauty. Arguably, he is talking about not only this particular woman, but complaining about the habits of women in general.
In the first stanza, the repetition of "still" has a cumulative effect; Jonson here uses enumeratio to set out the many rituals the woman undergoes: "still to be neat, still to be drest...Still to be powder'd, still perfum'd." The repetition builds to a climax, amplifying the sheer number of things that must be done, before the damning repetition of the final line: "All is not sweet, all is not sound." Here, the purpose of the repetition is to amplify the fact that such endless work has not had the desired effect: all is certainly not perfect.
Instead, Jonson sets out what he would prefer: "Give me...give me..." And yet the repetition in the second stanza occurs only once: Jonson does not wish to suggest that his own demands are excessive. Rather, the lack of repetition in the second half of the poem underlines the difference between its simplicity, echoing the simplicity of presentation Jonson calls for, and the busy overwork of the first stanza, reflecting the indulgence of effort focused on physical appearance.

How is Brown vs. Plata related to the Eighth Amendment? How did Escobedo vs. Illinois affect interrogations in the United States?

The 8th Amendment pertains to prison overcrowding and the Brown vs. Plata case. First, the 8th Amendment protects citizens against cruel and unusual punishments. In 2011, the Supreme Court affirmed that the prison conditions in the state of California violated the cruel and unusual punishment clause in the 8th Amendment. In a 5-4 decision, the justices ordered California to reduce its inmate population by at least 46,000 inmates. According to a New York Times article, extreme overcrowding in California prisons has resulted in administrative negligence and high prisoner suicide rates. Prisoners with mental and physical health issues often received little care for their conditions.
Here's a concise summary of the Brown vs. Plata case.
In regards to the sentencing of a defendant, lawyers and/ or prosecutors may highlight aggravating and mitigating factors in a bid to influence the judge's sentencing decision. Mitigating factors are used by the defendant's lawyers to plead for leniency. Some of these factors include the remorse of the defendant, the defendant's lack of a criminal record, the defendant's minor role in his crime, or the evidence of physical and/ or mental incapacity at the time of his crime.
On the other hand, prosecutors try to highlight aggravating factors that may lead to a harsher sentence for the defendant. Some of these factors include the severity of the victim's suffering, the level of the victim's vulnerability, the defendant's record of similar convictions, and the level of the defendant's participation in his crime.
In the case of interrogations, coerced confessions violate the Fifth Amendment right against self-incrimination. In Escobedo vs. Illinois, Escobedo was allegedly denied his Sixth Amendment right to counsel; therefore, he had no way of protecting his Fifth Amendment rights. Before the advent of the Miranda rule (where defendants are advised of their right to remain silent and/ or to have a lawyer present during an interrogation), the totality of circumstances test was used to determine whether a defendant's confession was in any way coerced (in violation of that person's Fifth Amendment rights).
In Rhode Island vs. Innis, the Supreme Court concluded that Thomas J. Innis's Fifth Amendment rights were not violated. Accordingly, Innis was arrested for committing a robbery with a sawed-off shotgun. At the time of his arrest, the shotgun was nowhere to be found. On the way to the police station, the officers openly admitted their fears that the shotgun might be found by children from a nearby school for the disabled. In response to this conversation, Innis offered to show the officers where the shotgun was. In a  6-3 decision, the Supreme Court ruled that the officers' conversation did not qualify as words that would elicit a self-incriminating confession from Innis.
In Quarles vs. NY, the Supreme Court ruled that Quarles had not been denied his Miranda rights. When Quarles was apprehended by an officer, the officer asked him about the location of his gun. After the officer received an answer from Quarles, he proceeded to read Quarles his Miranda rights. The court maintained that the officer's request constituted a public safety exception to the Miranda rule. Essentially, the officer had the right to protect his person and ensure the safety of bystanders before he read Quarles his Miranda rights.
https://constitution.findlaw.com/amendment5/annotation09.html

https://www.justia.com/criminal/aggravating-mitigating-factors/

int sec^4 (2x) dx Find the indefinite integral

 
Given to solve
int sec^4(2x)dx
let u= 2x
=> du = 2dx => dx = (1/2)du
so,
int sec^4(2x)dx
= int sec^4(u) (1/2) du
= (1/2) int sec^4(u) du
let us sovle
int sec^4(u) du
as by the formulae
int sec^n (x) dx
= (sec^(n-1) (x) (sinx))/(n-1) +((n-2)/(n-1))*(int sec^(n-2) (x) dx)
 
so,
int sec^4(u) du
=(sec^(4-1) (u) (sin u))/(4-1) +((4-2)/(4-1))*(int sec^(4-2) (u) du)
=(sec^(3) (u) (sin u))/(3) +(2/3)*(int sec^(2) (u) du)
= (sec^(3) (u) (sin u))/(3) +(2/3)*(tan u)
so,
int sec^4(2x)dx
= (1/2) int sec^4(u) du
=(1/2)[(sec^(3) (u) (sin u))/(3) +(2/3)*(tan u)]
but u= 2x
so,
(1/2)[(sec^(3) (u) (sin u))/(3) +(2/3)*(tan u)]
= (1/2)[(sec^(3) (2x) (sin (2x)))/(3) +(2/3)*(tan (2x))]
so,
int sec^4(2x)dx
=(1/2)[(sec^(3) (2x) (sin (2x)))/(3) +(2/3)*(tan (2x))] +c