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Finding a volume

Find the volume of the solid of revolution obtained by revolving the region enclosed by the graph of f ( x ) = x 2 ( x 2 + 1 ) 2 and the x -axis over the interval [ 0 , 1 ] about the y -axis.

Let’s begin by sketching the region to be revolved (see [link] ). From the sketch, we see that the shell method is a good choice for solving this problem.

This figure is the graph of the function f(x) = x^2/(x^2+1)^2. It is a curve above the x-axis. It is decreasing in the second quadrant, intersects at the origin, and increases in the first quadrant. Between x = 0 and x = 1, there is shaded area under the curve.
We can use the shell method to find the volume of revolution obtained by revolving the region shown about the y -axis.

The volume is given by

V = 2 π 0 1 x · x 2 ( x 2 + 1 ) 2 d x = 2 π 0 1 x 3 ( x 2 + 1 ) 2 d x .

Since deg ( ( x 2 + 1 ) 2 ) = 4 > 3 = deg ( x 3 ) , we can proceed with partial fraction decomposition. Note that ( x 2 + 1 ) 2 is a repeated irreducible quadratic. Using the decomposition described in the problem-solving strategy, we get

x 3 ( x 2 + 1 ) 2 = A x + B x 2 + 1 + C x + D ( x 2 + 1 ) 2 .

Finding a common denominator and equating the numerators gives

x 3 = ( A x + B ) ( x 2 + 1 ) + C x + D .

Solving, we obtain A = 1 , B = 0 , C = −1 , and D = 0 . Substituting back into the integral, we have

V = 2 π 0 1 x 3 ( x 2 + 1 ) 2 d x = 2 π 0 1 ( x x 2 + 1 x ( x 2 + 1 ) 2 ) d x = 2 π ( 1 2 ln ( x 2 + 1 ) + 1 2 · 1 x 2 + 1 ) | 0 1 = π ( ln 2 1 2 ) .
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Set up the partial fraction decomposition for x 2 + 3 x + 1 ( x + 2 ) ( x 3 ) 2 ( x 2 + 4 ) 2 d x .

x 2 + 3 x + 1 ( x + 2 ) ( x 3 ) 2 ( x 2 + 4 ) 2 = A x + 2 + B x 3 + C ( x 3 ) 2 + D x + E x 2 + 4 + F x + G ( x 2 + 4 ) 2

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Key concepts

  • Partial fraction decomposition is a technique used to break down a rational function into a sum of simple rational functions that can be integrated using previously learned techniques.
  • When applying partial fraction decomposition, we must make sure that the degree of the numerator is less than the degree of the denominator. If not, we need to perform long division before attempting partial fraction decomposition.
  • The form the decomposition takes depends on the type of factors in the denominator. The types of factors include nonrepeated linear factors, repeated linear factors, nonrepeated irreducible quadratic factors, and repeated irreducible quadratic factors.

Express the rational function as a sum or difference of two simpler rational expressions.

1 ( x 3 ) ( x 2 )

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x 2 + 1 x ( x + 1 ) ( x + 2 )

2 x + 1 + 5 2 ( x + 2 ) + 1 2 x

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3 x 2 x 2 + 1 ( Hint: Use long division first.)

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2 x 4 x 2 2 x

2 x 2 + 4 x + 8 + 16 x 2

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1 ( x 1 ) ( x 2 + 1 )

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1 x 2 ( x 1 )

1 x 2 1 x + 1 x 1

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1 x ( x 1 ) ( x 2 ) ( x 3 )

1 2 ( x 2 ) + 1 2 ( x 1 ) 1 6 x + 1 6 ( x 3 )

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1 x 4 1 = 1 ( x + 1 ) ( x 1 ) ( x 2 + 1 )

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3 x 2 x 3 1 = 3 x 2 ( x 1 ) ( x 2 + x + 1 )

1 x 1 + 2 x + 1 x 2 + x + 1

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3 x 4 + x 3 + 20 x 2 + 3 x + 31 ( x + 1 ) ( x 2 + 4 ) 2

2 x + 1 + x x 2 + 4 1 ( x 2 + 4 ) 2

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Use the method of partial fractions to evaluate each of the following integrals.

d x ( x 3 ) ( x 2 )

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3 x x 2 + 2 x 8 d x

ln | 2 x | + 2 ln | 4 + x | + C

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x x 2 4 d x

1 2 ln | 4 x 2 | + C

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d x x ( x 1 ) ( x 2 ) ( x 3 )

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2 x 2 + 4 x + 22 x 2 + 2 x + 10 d x

2 ( x + 1 3 arctan ( 1 + x 3 ) ) + C

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2 x x 2 + x d x

2 ln | x | 3 ln | 1 + x | + C

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2 x 2 x 6 d x

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d x x 3 2 x 2 4 x + 8

1 16 ( 4 −2 + x ln | −2 + x | + ln | 2 + x | ) + C

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d x x 4 10 x 2 + 9

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Evaluate the following integrals, which have irreducible quadratic factors.

2 ( x 4 ) ( x 2 + 2 x + 6 ) d x

1 30 ( −2 5 arctan [ 1 + x 5 ] + 2 ln | −4 + x | ln | 6 + 2 x + x 2 | ) + C

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x 2 x 3 x 2 + 4 x 4 d x

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x 3 + 6 x 2 + 3 x + 6 x 3 + 2 x 2 d x

3 x + 4 ln | x + 2 | + x + C

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x ( x 1 ) ( x 2 + 2 x + 2 ) 2 d x

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Use the method of partial fractions to evaluate the following integrals.

3 x + 4 ( x 2 + 4 ) ( 3 x ) d x

ln | 3 x | + 1 2 ln | x 2 + 4 | + C

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2 ( x + 2 ) 2 ( 2 x ) d x

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3 x + 4 x 3 2 x 4 d x ( Hint: Use the rational root theorem.)

ln | x 2 | 1 2 ln | x 2 + 2 x + 2 | + C

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Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.

0 1 e x 36 e 2 x d x (Give the exact answer and the decimal equivalent. Round to five decimal places.)

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e x d x e 2 x e x d x

x + ln | 1 e x | + C

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sin x d x 1 cos 2 x

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sin x cos 2 x + cos x 6 d x

1 5 ln | cos x + 3 cos x 2 | + C

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d t ( e t e t ) 2

1 2 2 e 2 t + C

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1 + e x 1 e x d x

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d x 1 + x + 1

2 1 + x 2 ln | 1 + 1 + x | + C

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cos x sin x ( 1 sin x ) d x

ln | sin x 1 sin x | + C

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e x ( e 2 x 4 ) 2 d x

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1 2 1 x 2 4 x 2 d x

3 4

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1 1 + e x d x

x ln ( 1 + e x ) + C

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Use the given substitution to convert the integral to an integral of a rational function, then evaluate.

1 t t 3 d t t = x 3

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1 x + x 3 d x ; x = u 6

6 x 1 / 6 3 x 1 / 3 + 2 x 6 ln ( 1 + x 1 / 6 ) + C

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Graph the curve y = x 1 + x over the interval [ 0 , 5 ] . Then, find the area of the region bounded by the curve, the x -axis, and the line x = 4 .

This figure is a graph of the function y = x/(1 + x). The graph is only in the first quadrant. It begins at the origin and increases into the first quadrant. The curve stops at x = 5.
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Find the volume of the solid generated when the region bounded by y = 1 / x ( 3 x ) , y = 0 , x = 1 , and x = 2 is revolved about the x- axis.

4 3 π arctanh [ 1 3 ] = 1 3 π ln 4

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The velocity of a particle moving along a line is a function of time given by v ( t ) = 88 t 2 t 2 + 1 . Find the distance that the particle has traveled after t = 5 sec.

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Solve the initial-value problem for x as a function of t.

( t 2 7 t + 12 ) d x d t = 1 , ( t > 4 , x ( 5 ) = 0 )

x = ln | t 3 | + ln | t 4 | + ln 2

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( t + 5 ) d x d t = x 2 + 1 , t > 5 , x ( 1 ) = tan 1

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( 2 t 3 2 t 2 + t 1 ) d x d t = 3 , x ( 2 ) = 0

x = ln | t 1 | 2 arctan ( 2 t ) 1 2 ln ( t 2 + 1 2 ) + 2 arctan ( 2 2 ) + 1 2 ln 4.5

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Find the x -coordinate of the centroid of the area bounded by

y ( x 2 9 ) = 1 , y = 0 , x = 4 , and x = 5 . (Round the answer to two decimal places.)

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Find the volume generated by revolving the area bounded by y = 1 x 3 + 7 x 2 + 6 x x = 1 , x = 7 , and y = 0 about the y -axis.

2 5 π ln 28 13

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Find the area bounded by y = x 12 x 2 8 x 20 , y = 0 , x = 2 , and x = 4 . (Round the answer to the nearest hundredth.)

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Evaluate the integral d x x 3 + 1 .

arctan [ −1 + 2 x 3 ] 3 + 1 3 ln | 1 + x | 1 6 ln | 1 x + x 2 | + C

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For the following problems, use the substitutions tan ( x 2 ) = t , d x = 2 1 + t 2 d t , sin x = 2 t 1 + t 2 , and cos x = 1 t 2 1 + t 2 .

Find the area under the curve y = 1 1 + sin x between x = 0 and x = π . (Assume the dimensions are in inches.)

2.0 in. 2

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Given tan ( x 2 ) = t , derive the formulas d x = 2 1 + t 2 d t , sin x = 2 t 1 + t 2 , and cos x = 1 t 2 1 + t 2 .

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Evaluate x 8 3 x d x .

3 ( −8 + x ) 1 / 3
−2 3 arctan [ −1 + ( −8 + x ) 1 / 3 3 ]
−2 ln [ 2 + ( −8 + x ) 1 / 3 ]
+ ln [ 4 2 ( −8 + x ) 1 / 3 + ( −8 + x ) 2 / 3 ] + C

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Questions & Answers

List and explain four factors of production
Vuyo Reply
capital labour entrepreneur natural resources
Thembi
What is supply
Ogodo Reply
when the supply decreases demand also decreases
Thembi
types of demand and the explanation
akin Reply
what is demand
akin Reply
other things remaining same if demend is increases supply is also decrease and if demend is decrease supply is also increases is called the demand
Mian
if the demand increase supply also increases
Thembi
you are wrong this is the law of demand and not the definition
Tarasum
Demand is the willingness of buy and ability to buy in a specific time period in specific place. Mian you are saying law of demand but not in proper way. you have to keep studying more. because its very basic things in Economics.
Hamza
Demand is the price of Quantity goods and services in which consumer's are willing and able to offer at a price in the market over a period of time
Umar
Demand is the quantity of goods and services that the consumer are willing and able to buy at a alternative prices over a given period of time. But mind you demand is quite different from need and want.
Tarasum
what is commercialization?
Doris Reply
How to talk loan for bank?
Alfred Reply
what is the meaning of gpa?
Ritisha Reply
Answer: GPA stands for Grade Point Average. It is a standard way of measuring academic achievement in the U.S. Basically, it goes as follows: Each course is given a certain number of "units" or "credits", depending on the content of the course.
Yusuf
what is small and Microbuisenes
tadesse Reply
What is fiscal policy
Dansofo
Who is the funder of Economic
Dansofo
founder , that is Adam Smith
Daniel
what is model
Daniel Reply
The wealth of Nations
Yusuf Reply
the wealth of nations, is it the first?
Umar
Yes very sure it was released in 1759
Yusuf
thank you Yusuf.
Umar
then when did he died?
Umar
17 July 1790 Born: 16 June 1723, Kirkcaldy, United Kingdom Place of death: Panmure House, Edinburgh, United Kingdom
Yusuf
1790
Yusuf
that's my today questions, thank you Yusuf it's bed time see u after.
Umar
what is fiscal policy
kemigisha Reply
what's mode?
Umar Reply
mode is the highest occurring frequency in a distribution
Bola
mode is the most commonly occurring item in a set of data.
Umar
Please, what is the difference between monopoly and monopsony?
Olaleye Reply
is there monopsony word?
Umar
I have no idea though
Umar
please, in which year Adam smith was born?
Umar
monopsony is when there's only one buyer while monopoly is when there's only one producer.
Bola
who have idea on Banter
Ibrahim
like trade by barter?
Bola
Monopoly is when there's excessively one seller and there is no entry in the market while monopsony is when there is one buyer
kemigisha
Adam smith was born in 1723
Bola
 (uncountable) Good humoured, playful, typically spontaneous conversation. verb (intransitive) To engage in banter or playful conversation. (intransitive) To play or do something amusing. (transitive) To tease mildly.
Umar
which book Adam smith published first? the first book of Adam smith pls.
Umar
wealth on nation, 1776
Daniel
what is market power and how can it affect an economy?
Gab Reply
market power:- where a firm is said to be a price setter.market power benefits the powerful at the expense of others.
Umar
Market power refers to the ability of a firm (or group of firms) to raise and maintain price above the level that would prevail under competition is referred to as market or monopoly power. The exercise of market power leads to reduced output and loss of economic welfare
Kartheek
find information about the national budget
Molahlegi
three branches of economics in which tourism is likely to figure
Makgotso Reply
What are those three branches?
IlRegno
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul
Practice Key Terms 1

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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