Answer:
[tex]\displaystyle \oint_C {(y - x) \, dx + (2x - y) \, dy} = \boxed{\bold{2 \pi}}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]:
[tex]\displaystyle \bold{(cu)' = cu'}[/tex]
Derivative Rule [Basic Power Rule]:
Integration
IntegralsIntegration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \bold{\int\limits^b_a {f(x)} \, dx = F(b) - F(a)}[/tex]
Integration Property [Multiplied Constant]:
[tex]\displaystyle \bold{\int {cf(x)} \, dx = c \int {f(x)} \, dx}[/tex]
Integration Property [Addition/Subtraction]:
[tex]\displaystyle \bold{\int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx}[/tex]
Integration Method: U-Substitution
Reduction Formula [Sine]:
[tex]\displaystyle \bold{\int {\sin^n x} \, dx = \frac{n - 1}{n} \int {\sin^{n - 2} x} \, dx - \frac{\cos x \sin^{n - 1} x}{n} + C}[/tex]
Reduction Formula [Cosine]:
[tex]\displaystyle \bold{\int {\cos^n x} \, dx = \frac{n - 1}{n} \int {\cos^{n - 2} x} \, dx + \frac{\cos^{n - 1} x \sin x}{n} + C}[/tex]
Multivariable Calculus
Partial Derivatives
Partial Derivative Rule [Chain Rule]:
[tex]\displaystyle \bold{\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial t}}[/tex]
Vector Calculus
Circulation Density:
[tex]\displaystyle F = M \hat{\i} + N \hat{\j} \rightarrow \text{curl} \ \bold{F} \cdot \bold{k} = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}[/tex]
Green's Theorem [Circulation Curl/Tangential Form]:
[tex]\displaystyle \bold{\oint_C {F \cdot T} \, ds = \oint_C {M \, dx + N \, dy} = \iint_R {\bigg( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \bigg)} \, dx \, dy}[/tex]
Step-by-step explanation:
Step 1: Define
Identify given.
[tex]\displaystyle \oint_C {(y - x) \, dx + (2x - y) \, dy}[/tex]
[tex]\displaystyle \text{Parametrized by:} \left\{\begin{array}{ccc} x = 2 \cos t \\ y = \sin t \\ 0 \leq \theta \leq 2 \pi \end{array}[/tex]
Step 2: Integrate Pt. 1
Parametrize integrand:Step 3: Integrate Pt. 2
We can evaluate the Green's Theorem double integral we found using basic integration techniques listed above:
[tex]\displaystyle \begin{aligned}\oint_C {(y - x) \, dx + (2x - y) \, dy} & = \int\limits^{2 \pi}_0 {-2 \sin t(\sin t - 2 \cos t)+ \cos t(4 \cos t - \sin t)} \, dt \\& = \int\limits^{2 \pi}_0 {-2 \sin t(\sin t - 2 \cos t)} \, dt + \int\limits^{2 \pi}_0 {\cos t(4 \cos t - \sin t)} \, dt \\& = 4 \int\limits^{2 \pi}_0 {\cos t \sin t - 2 \sin^2t} \, dt + 4 \int\limits^{2 \pi}_0 {\cos^2t - \cos t \sin t} \, dt \\\end{aligned}[/tex]
[tex]\displaystyle\begin{aligned}\oint_C {(y - x) \, dx + (2x - y) \, dy} & = 4 \int\limits^{2 \pi}_0 {\cos t \sin t} \, dt - 2 \int\limits^{2 \pi}_0 {\sin^2t} \, dt + 4 \int\limits^{2 \pi}_0 {\cos^2 t} \, dt - \int\limits^{2 \pi}_0 {\cos t \sin t} \, dt \\& = \bigg( 2 \sin^2t + \cos t \sin t - t \bigg) \bigg| \limits^{2 \pi}_0 + \bigg( \sin 2t + \frac{\cos 2t}{4} + 2t \bigg) \bigg| \limits^{2 \pi}_0 \\& = -2 \pi + 4 \pi \\& = \boxed{\bold{2 \pi}}\end{aligned}[/tex]
∴ we have evaluated the line integral using Green's Theorem.
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Topic: Multivariable Calculus
Unit: Green's Theorem and Surfaces
what was one cause of the decline of the Incan Empire?
A. continual war with the aztec empire
B. drought caused by climate change
C. introduction of diseases from Spanish explorers
D. destruction of natural resources
Answer:
c. introduction of diseases from spanish explorers
Explanation:
It said that in the textbook and i just got it right on the test
Answer:introduction of diseases from spanish explorers
Explanation:
PENN
Which factor most contributed to the development of trench warfare?
A. New tactics favored offense and infantry charges.
B. Weapons like machine guns could easily kill troops in the open.
C. Germany couldn't stop British and French advances, so German
troops dug trenches to delay the attack.
D. Leaders felt that digging trenches would lead to a quick war of
movement.
as a consequence for his actions during the red scare, what happened to joseph mccarthy
Answer:
On December 2, 1954, the Senate voted to censure Senator McCarthy by a vote of 67–22, making him one of the few senators ever to be disciplined in this fashion. He continued to speak against communism and socialism until his death at the age of 48 at Bethesda Naval Hospital in Bethesda, Maryland, on May 2, 1957
Explanation:
~Hope this helps
please make sure you’re right i this is for a growth measure
How were the motivations of the Spanish Empire in the 16th century and the British Empire in the 19th century similar?
A. They both wished to subjugate indigenous people in their territories to bring more people under their monarch’s control.
B. They both wanted to use their territories as sources of raw goods for increased wealth and economic power.
C. They both sought to Christianize the territories under their control by forcing native peoples to convert.
D. They both sought to make political alliances by creating colonies that would become allies throughout the world.
Answer: B. They both wanted to use their territories as sources of raw goods for increased wealth and economic power.
Explanation:
Both the British and the Spanish viewed their colonies as a means to get raw materials cheaply so that they may increase the wealth of their respective nations which would also lead to increased economic power.
The British took advantage of the various industries on offer in the American colonies from ship building to plantation agriculture to increase their wealth and the Spanish exploited their colonies for agriculture and minerals.
King Charles V of Spain gained control of which other empire in 1519?
O A. Ottoman empire
O B. Habsburg empire
O C. British empire
OD. Portuguese empire
Answer:
spain empire is the answer
what are examples of personal rights
Answer:
The right to live your life free of discrimination. The right to control what happens to your own body and to make medical decisions for yourself. The right to freely exercise your religion and practice your religious beliefs without fear of being prosecuted for your beliefs.
Explanation:
When did the Vikings place a settlement in North America?
Answer:late 10th century CE
Explanation:In the 10th Century, the Viking world stamped it’s foot on Europe
pleaseee
Which of the following is not a part of the Georgia justice system?
city council
juvenile court
law enforcement
court system
Which of the following is an example of government influence on supply? A) law of supply B) excess demand C) marginal costs D) market supply curve
Answer:
the answer I A) law of supply
3. Which of the following best explains the purpose of propaganda? 100pts
Answer:
where's the following?
Answer:
Examples of propaganda of the deed would include staging an atomic “test” or the public torture of a criminal for its presumable deterrent effect on others, or giving foreign “economic aid” primarily to influence the recipient's opinions or actions and without much intention of building up the recipient's economy.
The bill of right would best represent which prinicple of government? A. Individual right b. Popular sovereignity C. Limited government D. Separation of power
Answer:
Option A, Individual rights
Hope it helps!
Help ASAP please asappp
Which answer best describes President Johnson's role in Reconstruction?
Group of answer choices
He refused to take the presidential oath of office.
He vetoed most Reconstruction laws that would increase the rights of the freed slaves
He withdrew the military from Southern states.
Answer:
He vetoed most reconstruction laws that would increase the rights of freed slaves
ExplaJohnation:
Read these lines from the poem.
Yet here it's still the school year, and it can't end too soon, / But the days are standing still, stretching till the end of June.
What does the author mean by the phrase the days are standing still ?
Group of answer choices
The days are full of difficult work.
There are more school days left than the speaker thought.
The days seem to pass by slowly.
The speaker's school day is longer than at most schools.
Answer:
hoummmmmmmmExplanation:
how read?
In the mid-twentieth century, which of the following was a similarity between the approaches of China and the Soviet Union in managing their respective economies?
Answer:
Insistence on the participation of industrial workers is planning their economies
What was not a trade region connected to Constantinople ?
Answer: Byzantine empire
Explanation:
What is preamble exactly? Please go into lots of detail!
Answer:
It is an introductory of sorts to documents.
Explanation:
A preamble is an introductory and expressionary statement in a document that explains the document's purpose and underlying philosophy. When applied to the opening paragraphs of a statute, it may recite historical facts pertinent to the subject of the statute. It is distinct from the long title or enacting formula of a law.
write an essay on chola administration and their contribution to art and architecture.
Answer:
Chola had a great contribution to art and architecture.
Explanation:
Chola Empire was divided into three major administrative i.e. Central Government, Provincial government and local government. This division was done in order to make easiness in the government. Chola had a great contribution to art and architecture by building temples in South India. The Cholas were great patrons of art, during their reign, the most magnificent temples and exquisite bronze icons were built in South India. Siva Temple of Thanjore which is the largest and tallest Indian temple was built in Chola Period.
why are dhows so important in the history of asia and africa?
answer:
dhows are so important because they helped with the trade. they are trading vessels used to carry heavy items.
explanation:
although there were many deep-water ships, the dhows were predominantly used for coastal trade moving up and down the east african coast, the dhows stopped at ports along the way, trading goods and ferrying passengerscredits: online researchWhich statement best describes how Caesar Augustus helped the Roman Empire expand?
1. He reduced military spending and saved the empire a lot of money.
2. He let his generals lead the army and make military decisions.
3. He improved the army and the roads, and created a navy.
4. He negotiated peace deals with his neighbors.
Answer:
He improved the army and the roads, and created a navy
Explanation:
Order the fractions from least to greatest.
2 4 7 2
4' 5' 10'3
A)
2 7 2 4
4' 10'3'5
B)
7 2 2 4
10'4' 3'5
C)
2 2 7 4
4' 3' 10'5
D)
2 7 4 2
4' 10'5'3
In Congress, who has the right to unlimited debate and unlimited opportunity to offer amendments?
A.
the president
B.
senators
C.
the Speaker of the House
D.
the vice president
E.
majority leaders
The graph shows a supply curve.
Sample Supply Curve
S1
S
Price
$20.00
$17.50
$15.00
$12.50
$10.00
$750
55.00
S250
0
30
10
20
Quantity Supplied
Which change is illustrated by the shift taking place on this graph?
O an increase in demand
O a decrease in demand
an increase in supply
a decrease in supply
The change illustrated in the graph is a decrease in supply.
What is the supply curve?
The supply curve is a positively sloped curve that shows how quantity supplied changes with price of the good. All things being equal, the higher the price of the good, the higher the quantity supplied.
What is a decrease in supply?
A decrease in supply is when the supply curve shifts to the left. Factors that cause a decrease in supply are:
an increase in the price of inputsa decrease in the number of suppliers.To learn more about supply curves, please check: https://brainly.com/question/26073189
a.
Which of the following statements regarding the election of 1860 is FALSE ?
The Democratic Party was split along sectional lines.
b. Northern Democrats nominated Abraham Lincoln for the Presidency.
C. The Republican Party formed after the Kansas-Nebraska Act.
d. The Constitutional Union Party supported neither Democrats or Republicans.
Answer:
C. The Republican Party formed after the Kansas-Nebraska Act.
Explanation:
I got it right on plato.
Please label the map the labels is at the bottom left
Answer: B is pacific ocean
A is papua new guinea
C is great barrier reef.
D is outback
E is perth
F is sydney
H is north island
G is melbourne
sprry for them being out of order
:)
how did the samurai learn how to use weapons? how did the samurai use there weapons when they were in battle?
Answer: They watched CoryxKenshin
Explanation:
How were taxes handled with the Articles of Confederation?
Answer:
One of the biggest problems was that the national government had no power to impose taxes. To avoid any perception of “taxation without representation,” the Articles of Confederation allowed only state governments to levy taxes. To pay for its expenses, the national government had to request money from the states.
Explanation:
How did the African Trading Kingdoms' successes lead to their eventual demise at the hands of the Europeans?
Answer:
Ghana's empire reached its height around the year 1000 C.E. War and the loss of natural resources led to the West African empire's downfall, and the rise of a new power. In the second half of the 11th century, Muslim warriors known as Almoravids began attacking Ghana's empire.
Explanation:
what methods did the Nazis use to separate Jewish people and other targeted groups from the rest of the population?
The Nazis separated Jewish people into small, condensed communities called ghettos. They also gave the Jewish populace Yellow patches (or badges), so, they would be able to identify the Jews from the rest of the population.