Find the centroid of the region bounded by the graphs of `y=sqrt(r^2-x^2)` and `y=0`

Given the curves are

`y=sqrt(r^2-x^2)` (this is a circle equation actually, but lets compute the centroid  in a standard way) ,` y=0`

let `f(x) =sqrt(r^2-x^2)`

and `g(x)=0`

In order to find the Centroid of the region bounded by the curves,

first we have to find the area bounded by the curves.

So now in order to find the area , we have to find the intersecting points of the curves. This can be obtained by equating f(x) and g(x) .

=>` f(x) = g(x)`

=> `sqrt(r^2-x^2) =0`

=>` (r^2-x^2) =0`

=> `x^2 = r^2`

=> `x=+-r` --------(1)

so the curves `f(x)>=g(x) , -r<=x<=r`

so the area `= int _-r ^r [sqrt(r^2 - x^2) -0]dx`

let `x=rsin(theta)` ------(2)

so , `dx = rcos(theta) d theta`

but from (1) and (2) we get

`x=+-r , x= rsin(theta)` 

So,

`sin(theta) = +-1`

=> `theta = sin^(-1) (+-1)`

so` theta = +-(pi/2)`

so, now with the new integrals we get

area` = int _-r ^r [sqrt(r^2 - x^2) -0] dx`

  = `int_(-pi/2) ^(pi/2) sqrt(r^2 - r^2sin^2 (theta)) rcos(theta) d theta`

  = `int_(-pi/2) ^(pi/2) sqrt(r^2(1 - sin^2 (theta))) rcos(theta) d theta` 

  =`int_(-pi/2) ^(pi/2) sqrt(r^2( cos^2 (theta))) rcos(theta) d theta`

  =`int_(-pi/2) ^(pi/2) (rcos (theta)) rcos(theta) d theta`

  =`int_(-pi/2) ^(pi/2) (rcos (theta))^2 d(theta)`

  =`(r^2)int_(-pi/2) ^(pi/2) (cos^2 (theta)) d(theta)`

we can right the above integral as

  =`(2) (r^2)int_0 ^(pi/2) (cos^2 (theta)) d(theta)`

  as we know that `int cos^2(x) dx = (1/2)(x+(1/2)sin (2x))`

now ,

area = `(2) (r^2)int_0 ^(pi/2) (cos^2 (theta)) d(theta)`

=`(2) (r^2) [(1/2)(x+(1/2)sin (2x))]_0 ^(pi/2)`

 = `2(r^2)[((1/2)((pi/2)+(1/2)sin (2(pi/2))))-(1/2)((0)+(1/2)sin (2(0)))]`

=`2(r^2)[((1/2)((pi/2)+(0)))-0]`

=`2 r^2[pi/4]`

=` (pi r^2)/2`

Now the centroid of the region bounded by the curves is given as 

let `(x_1,y_1)` be the coordinates of the centroid so ,

`x_1` is given as

`x_1 = (1/(area)) int _a^b x[f(x)-g(x)] dx`

  where the `a= -r , b= r`

  = `(1/((pi r^2)/2)) int _-r ^r x[sqrt(r^2 -x^2)-(0)] dx`

  =`(1/((pi r^2)/2)) int _-r ^r x[sqrt(r^2 -x^2)] dx`

  =` (2/(pi r^2)) int _-r ^r [sqrt(r^2 -x^2)] xdx`

let` u = r^2 -x^2 => du = -2x dx`

`(-1/2)du = xdx`

So the bounds of integration are from `u=0` to `u=0` (plug in `x=-r` and `x=r` to `u=r^2-x^2` ) 

= `(2/(pi r^2)) int _-r ^r [sqrt(r^2 -x^2)] xdx`

= `(2/(pi r^2)) int _0 ^0 [sqrt(u)] (-1/2)du`

= `0` as `int_a^b f(t) dt = F(a) - F(b)` where F(x) is the antiderivative of f. So, `int_0^0 f(t) dt = F(0)-F(0)=0`

so, `x_1 = 0`

and now let us `y_1` and so ,

`y_1` is given as

`y_1 = (1/(area)) int _a^b (1/2) [f^2(x)-g^2(x)] dx`

    where `a= -r , b= r`

= `(1/((pi r^2)/2)) int _-r ^r (1/2)[(sqrt(r^2 -x^2))^2] dx`

= `(2/((pi r^2))) (1/2) int _-r ^r [(sqrt(r^2 -x^2))^2] dx`

 = `(1/((pi r^2))) int _-r ^r [((r^2 -x^2))] dx`

Because `r^2-x^2` is an even function with respect to x, we can change the integral limits to

 =`(2/((pi r^2))) int _0 ^r [((r^2 -x^2))] dx`

 = `(2/((pi r^2))) ((r^2)x -x^3 /3)_0 ^r `

 =` (2/((pi r^2))) [ [(r^3 -r^3 /3)]-[((r^2)0 -0^3 /3)]]`

= `2/(pi r^2) [(r^3 -r^3 /3)-0]`

= `(4r)/(3pi)`

Therefore, the centroid coordinates are `(0,(4r)/(3pi))`

What makes a democracy unique compared to other forms of government?

A democracy is unique in that the people rule it by popular vote. This gives everyone in society, regardless of class, the same amount of political power. Contrast this with a monarchy or a dictatorship (where one person rules according to his or her whim) or an oligarchy (where a small group of people can make the major decisions for a country). 

Some would argue that democracies are the best form of government in that...

A democracy is unique in that the people rule it by popular vote. This gives everyone in society, regardless of class, the same amount of political power. Contrast this with a monarchy or a dictatorship (where one person rules according to his or her whim) or an oligarchy (where a small group of people can make the major decisions for a country). 

Some would argue that democracies are the best form of government in that the people, if taken as a group, will make the best decisions for themselves. They will not go to war unless its to ensure the nation's survival—people are not willing to vote themselves into a position where they may die. Also, the people will insist on efficient governance with minimal taxes because people do not vote for their property to be taken away without a good reason. Detractors of democracies claim that the people can be selfish and are often swayed by whoever is the best speaker at the time, regardless of his or her qualifications for office. Others claim that democracies are slow in that one has to get a vote to get anything accomplished; given the size of the population, this vote can be hard to acquire. Still, others claim that a democracy can crush the rights of the minority, thus isolating an entire group of people from the political process.  

Explain the relationship between the limits to presidential power and the importance of the president's informal powers.

Presidential Powers

Before we discuss limits on president power and how that connects to informal powers, it is helpful to understand what the president can do, according to the Constitution:

• The President can execute (carry out) the laws made by Congress


• The President is commander in chief of the Army and Navy of the United States


• The President has the power to grant pardons for offenses, except in cases of impeachment


• The President can make treaties with 2/3rds consent of the Senate


• The President can nominate ambassadors, Supreme Court justices, cabinet officers and other executive officers with consent of the Senate

Limits to Presidential Power

The President's powers are limited by the system of checks and balances present in our government.  This system is also written into the Constitution.  The Legislative branch can overrule a Presidential veto with a 2/3rds vote, stop budgeting, and impeach the President.  The Judicial branch cannot be tampered with; once appointed, judges sit in their position for life.  The Courts can also use judicial review, which examines actions by the Legislature or Executive branch for constitutionality.  If the Courts find an action unconstitutional, they can reverse it.

Connections to Informal Powers

The President's informal powers are:

• Setting the agenda for Congress


• Issuing executive orders


• Sending out troops without a declaration of war


• Conducting initiatives and actions concerning foreign policy

How are these informal powers connected to the limits and restrictions on formal Presidential power?  

• Since the Legislative branch has the power to override and veto Presidential action, the President can use his informal power of setting the agenda wisely- if the agenda does not contain any hot button issues, or if the President waits until he has a Congress that shares his partisan views, he can assume that his actions will be accepted or passed.  It requires a 2/3rds vote in both the House and Senate to override Presidential action.  


• The President has the power to issue executive orders, a rule or law issued DIRECTLY from the President without going through Congress.  This bypasses the lengthy process of turning a bill into law.  However, it is important to note that is the executive order is unconstitutional, then the Supreme Court can overrule it.  


• The President has the ability to send troops abroad without waiting for the Senate to formally declare war.  This has been used several times in the recent past with the understanding that immediate action needed to take place before the Senate could vote on declaring war, like when George W. Bush sent troops to Afghanistan in the wake of the 9/11 terror attacks.

The President's information powers are a way to bypass some of the checks on Executive power, but they do not give the President free reign as head of the country.

How do Scout and Jem meet Dill in To Kill a Mockingbird?

In Chapter 1, Scout and Jem are playing in their backyard when they hear something coming from Miss Rachel Haverford's collard patch. They walk over to the fence expecting to find a puppy, and instead discover a small boy sitting down, staring directly at them. The boy introduces himself as Charles Baker Harris and mentions that he can read. When Jem asks Dill how old he is, Jem is surprised to learn that Dill is...

In Chapter 1, Scout and Jem are playing in their backyard when they hear something coming from Miss Rachel Haverford's collard patch. They walk over to the fence expecting to find a puppy, and instead discover a small boy sitting down, staring directly at them. The boy introduces himself as Charles Baker Harris and mentions that he can read. When Jem asks Dill how old he is, Jem is surprised to learn that Dill is seven years old. Jem initially thought Dill was four-and-a-half because he was so small. Dill says he is from Meridian, Mississippi and is staying with his aunt, Miss Rachel Haverford, for the summer. After Dill tells Jem that he's seen the movie Dracula, Jem begins to respect him. Jem and Scout quickly become close friends with Dill, and the three children play together every day during the summer. 

What is the definition for bad health as a result of a lack of food or eating unhealthy food?

Bad health resulting from a lack of food could be termed 'dietary deficiency,' 'malnutrition,' or 'lack of nutrients.' Severe lack of food could be starvation, whereas mild lack of food could be hunger or eating meals slightly less regularly.

Bad health resulting from eating unhealthy food says very little at all except that there is a claim that the unhealthy food is either partly or fully to blame for whatever health problem is apparent from...

Bad health resulting from a lack of food could be termed 'dietary deficiency,' 'malnutrition,' or 'lack of nutrients.' Severe lack of food could be starvation, whereas mild lack of food could be hunger or eating meals slightly less regularly.

Bad health resulting from eating unhealthy food says very little at all except that there is a claim that the unhealthy food is either partly or fully to blame for whatever health problem is apparent from symptoms.

Deficiencies in well-known vitamins (including A, B, C, D, and K) are associated with well-known symptoms, which is why those biochemical substances are termed vitamins. 

'Unhealthy food' is a modern parlance. The older term 'malnutrition' spoke of lack of basic vitamins, arising from eating only scraps and not being properly cared for by society. 'Unhealthy food' results from having too many choices about what to eat. People with the means to eat whatever food they choose can still choose 'unhealthy food.' Better-off members of society often fault chronic diseases and other health issues to diet. Cardiovascular disease, diabetes, and allergies are some examples. Scientific research is ongoing as to whether 'nutrition' (establishing what is and what isn't 'healthy food') can significantly help tackle chronic diseases, and also slow progressive diseases. Acute disease caused by 'unhealthy food' would mean the food is dangerously toxic or poisonous, beyond a level tolerable by the body. 'Unhealthy food' of that nature would be much better described as simply poison.

Why is Charlemagne remembered in history?

Charlemagne, also called Charles the Great, was a Frankish King who is widely regarded as the father of modern Europe. In addition to being a king, he united much of the territory of (present-day) Europe under the Holy Roman Empire.

In the Middle Ages, it was not uncommon for warlord kings to gain great followings and capture neighboring kingdoms to rule for themselves. When Charlemagne's father, Pippin the Short, died, he split the Frankish kingdom between...

Charlemagne, also called Charles the Great, was a Frankish King who is widely regarded as the father of modern Europe. In addition to being a king, he united much of the territory of (present-day) Europe under the Holy Roman Empire.

In the Middle Ages, it was not uncommon for warlord kings to gain great followings and capture neighboring kingdoms to rule for themselves. When Charlemagne's father, Pippin the Short, died, he split the Frankish kingdom between his two sons- Charlemagne and Carloman. Later, when Carloman died, Charlemagne ignored the rights of his heirs and took over his deceased brother's kingdom as ruler of all Francia. From then on, Charlemagne and his military began expanding and capturing territories belonging to the Saxons, the Gauls, and the Lombards all over the continent. 

Though he had captured so much territory and instituted great Administrative reforms all throughout his lands, Charlemagne wasn't officially recognized by the Church until the year 800. Something important to bear in mind is that from the fall of the Roman Empire in the West during the 4th century, the Church and the Crown were separate entities. In the eyes of the people, both were somewhat weakened without the support of the other. In November of 800, Charlemagne traveled to Rome to meet with Pope Leo III, and was crowned Emperor of the Romans. With the Church and Crown reunited, the Holy Roman Empire wielded both secular and religious authority. 

In addition to his Administrative reform throughout the Empire, Charlemagne inspired religious reform and implemented academic reform. One of his longest-lasting contributions to Western culture was introducing the Caroline Miniscule- a standardized script for all religious and academic texts to be written in. 

Based on lines 1-4 of "Remember," what can you conclude about the person being addressed by the speaker?

Well, this is an interesting question, especially stopping there. Start by being very clear: the poet does not say much directly about the person being addressed. It is all implied or communicated through the speaker discussing herself. Start with the first line—"Remember me when I am gone away." The person addressed is close enough to the speaker that she can ask him or her to remember after departure. The person addressed is also going to...

Well, this is an interesting question, especially stopping there. Start by being very clear: the poet does not say much directly about the person being addressed. It is all implied or communicated through the speaker discussing herself. Start with the first line—"Remember me when I am gone away." The person addressed is close enough to the speaker that she can ask him or her to remember after departure. The person addressed is also going to stay when the speaker leaves.

Go to the second line: "Gone far way into the silent land." The speaker is dying, or planning for death, and by implication, the person addressed is not. The person being addressed is either not ill or not old.

The third line reads, "When you can no more hold me by the hand." You would only say this to someone who took you by the hand, and repeatedly. These people are close. They might be lovers; they certainly love one another.

The final line "Nor I half turn to go yet turning stay" underscores this relationship. The speaker is close to the person addressed, drawn to them, and loves them.