Find the area bounded by y=8-x^2 and y=x^2.

First, we'll have to identify the limits of integration.
For this rason, we'll determine the intercepting points of the
graphs.

Since y = x^2 and y = 8-x^2, we'll
get:

x^2 = 8-x^2

We'll move
all terms to one side:

x^2 - 8 + x^2 =
0

2x^2 - 8 = 0

We'll divide by
2:

x^2 - 4 = 0

x^2 =
4

x1 = 2 and x2 = -2

Over the
interval [-2 ; 2] the graph 8-x^2 >= x^2

The area of
the region is the definite integral of the function 8 - x^2 - x^2, for x = -2 to x =
2.

A = Int (8 - x^2 - x^2)dx

A
= Int (8 - 2x^2)dx

A = Int 8dx -  Int
2x^2dx

A = 8x - 2x^3/3, for x = -2 to x =
2

A = 8(2 + 2) - (2/3)(8 +
8)

A = 32 - 32/3

A =
(96-32)/3

A = 64/3 square
units.

The area is A =  64/3 square
units.

What is the plot of "Soldier's Home"?

The plot of "Soldier’s Home" involves Krebs as the
protagonist in conflict with the abstract antagonism of peacetime adjustment. Obviously,
those around him expect him to be the little boy he was, and to be unchanged by the war.
They therefore assume that he can begin life again as though nothing has happened. The
crisis of the story is his conversation with his mother, and the climax is his promise
to be a good boy for his "Mummy" (paragraph 86). This portion of the story indicates
that his disaffection is approaching total alienation. The resolution, in his decision
to go to Kansas City and in his going to watch his sister play ball, suggests a
compromise with his disaffection. Though he will continue to feel like an outsider, in
other words, he will keep his misgivings to himself, and will fit in, at least
externally, with life at home.

What is the significance of Italy's invasion of Libya and Greece?

Well, to clarify, Libya was already an Italian colony at
the beginning of World War II, so Italy didn't need to invade it.  You may be thinking
of Ethiopia, which Mussolini attacked in 1936.  While he was successful at conquering
this poor, ill-equipped nation, it demonstrated that his army was no
juggernaut.

He fared even worse in Greece, where a massive
Italian attack was stopped by a few understrength divisions of the Greek Army.  For six
months, Mussolini's forces failed to gain the upper hand.  He called his friend Hitler,
who agreed to help him out by invading both Yugoslavia and Greece in "Operation
Punishment".  German forces conquered both nations in a mere seven
days.

This operation delayed the attack on the Soviet Union
by five weeks, which would become crucial the following December, when German troops
stalled right outside of Moscow as winter set in.  If they had attacked in early May
instead, as planned, they likely would have taken the city.

What are the solutions of the equation: cos x + cos 3x = 0?

We'll transform the sum in product using the
formula:

cos a + cos b = 2 cos[(a+b)/2]cos
[(a-b)/2]

We'll write the formula for a = x and b =
3x:

cos x + cos 3x = 2cos[(x+3x)/2]cos
[(x-3x)/2]

cos x + cos 3x = 2cos[(4x)/2]cos
[(-2x)/2]

cos x + cos 3x = 2cos[(2x)]cos
[(-x)]

Since the function cosine is even, we'll write cos
[(-x)] = cos x.

cos x + cos 3x = 2cos 2x*cos
x

We'll put 2cos 2x*cos x =
0

cos 2x = 0

2x = +/- arccos 0
+ 2kpi

2x = +/-pi/2 +
2kpi

x = +/- pi/4 +
kpi

cos x =
0

x = +/- pi/2 +
2kpi

The solutions of the
equation are: {+/- pi/4 + kpi}U{+/- pi/2 + 2kpi}.

Why have the Democratic and Republican parties been so durable so as to maintain exsistence since the Civil War? Explain.

I think pohnpei's response is right on.  Ours has been
designed as a two party system, first by the Founding Fathers in order to avoid what
Washington called "The dangers of faction".  Since that time, the two party system
supports and continues itself, making it very difficult for other interested parties to
enter the race.

We should acknowledge that there are many
more than two parties in the US today.  The Green Party, Socialist Party, Communist
Party and Reform Party to name a few.  These parties always put up candidates for the
Presidency, and always lose badly.  They are not even allowed to attend
debates.

The Democrat and Republican parties have loads of
cash, name recognition, a base of support, and a hammerlock on power.  They also try to
absorb other political movements and would-be parties by including some of the issues
important to them in their party "platforms" adopted at the national
conventions.

What are the literary devices used in the poem "The Masked Face" by Thomas Hardy?I want to know all the literary devices used in this poem such as...

Although this poem contains no examples of similes,
metaphors and personification, you would do well to pay attention to the other sound
effects that Hardy uses in this poem. Chief among these is alliteration, which appears a
number of times in this short poem. Consider the following examples and what the
alliteration does to the description. Firstly, the first line begins: "I found in me a
great surging space..." Note how the alliteration in the "s" sound helps emphasise the
size and bewildering state of this space that the speaker finds within himself.
Secondly, the speaker says that this space has no "firm-fixed floor," which again
emphasises the disturbing nature of this environment. Consider how the repetition of the
"f" sound is used again in the second stanza, as the doors are described as "fast
locked, and fill with fear." The "f" sound here seems to help convey the fear of the
narrator when confronted with these portals that entrap him in this "giddying
space."

Lastly, and key to the meaning of the poem, the
masked face uses an analogy in the last stanza that effectively compares our life and
the way we often complain about our situation to a pen that complains about what it is
being used to write because it doesn't understand
it.

Therefore, although this poem is absent of figurative
language such as similes and personification, there are ample examples of alliteration
that help convey the mood of fear and bewilderment of the speaker and also an analogy
which is used to convey the theme of the poem.

A parallelogram is formed in R^3 by the vectors PA=(3, 2, -3) and PB=(4, 1, 5). The point P=(0, 2, 3)Find the area of the parallelogram.

One side of the parallelogram is the vector PA and another
is PB. Let the fourth vertex be C.

The vertex A is (0, 2,
3) + (3, 2, -3) = (3, 4, 0)

The vertex B is (0, 2, 3) + (4,
1, 5) = (4, 3, 8)

The fourth vertex C is A + PB = (7, 5,
5)

Now, the diagonals of the parallelogram
are:

AB = (3, 4, 0) - (4, 3, 8) = (-1, 1, -8). |AB| = sqrt
(1+1+64) = sqrt(66)

PC = (7, 5, 5) - (0, 2, 3) = (7, 3, 2),
|PQ| = sqrt(49+9+4) = sqrt(62)

Let the angle APB be equal
to x, this gives:

|AB|^2 = |PA|^2 + |PB|^2 - 2 |PA| |PB|
cos x

=> 66 = 22 + 42 - 2*sqrt(22*42)*cos
x

=> cos x =
1/sqrt(22*42)

=> sin x = sqrt ( 1 - 1/(sqrt 22*42)^2
)

=> sin x = sqrt (22*42 -
1)/sqrt(22*42)

Taking an altitude from vertex A for the
triangle PAB, we have

h = |PA| sin x = sqrt(22) sqrt
(22*42-1) / sqrt(22*42)

The area of the triangle is
(1/2)*|PB|*h. The parallelogram is twice this area or the required area
is:

2*(1/2)*|PB|*h

=>
sqrt(42)*sqrt(22)*sqrt (22*42 - 1)/sqrt(22*42)

=>
sqrt(22*42 - 1)

=>
sqrt(923)

=>30.38

The
required area of the parallelogram is 30.38