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#1 2006-01-13 15:57:36

hristo
Member
Registered: 2005-12-26
Posts: 8

calculus, volume

hi,

i have a problem with a volume question. it's the following:

Let R be the shaded region in quadrant 1  bounded by the graphs of y = ln x and y = 1.
find the volume of the solid generated by rotating R about the y-axis.

book-answer: pi/2 (e^2-1)

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#2 2006-01-13 16:18:43

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

Re: calculus, volume

y = ln x



Draw a graph and note are going to integrate from
y = 0 to y = 1.

because of pi r^2 for area.  (disk method, might be called)

From an integral table:


Great answer, but I wish I knew how to
do the integral without the table lookup.

Last edited by John E. Franklin (2006-01-13 16:33:02)


igloo myrtilles fourmis

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#3 2006-01-13 16:36:10

hristo
Member
Registered: 2005-12-26
Posts: 8

Re: calculus, volume

but what happens with the y = 1? i need it to find the limits. it can't be the upper limit because the graphs don't intersect there. they intersect when x = e in this equation: y = ln x, so the upper limit should be e. correct me if i am wrong and please tell me why i am wrong.

Last edited by hristo (2006-01-13 16:43:47)

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#4 2006-01-13 16:50:01

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

Re: calculus, volume

Click on graph to make bigger.

View Image: lnx&y1.jpeg

igloo myrtilles fourmis

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#5 2006-01-13 16:51:12

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

Re: calculus, volume

I am integrating along the y-axis, not the x-axis.


igloo myrtilles fourmis

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#6 2006-01-13 16:54:22

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

Re: calculus, volume

For each microscopic y position, imagine a thin disc
that goes around the y-axis, so the discs are all
stacked up like coins if y is up and x is to the right.


igloo myrtilles fourmis

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#7 2006-01-13 16:55:49

hristo
Member
Registered: 2005-12-26
Posts: 8

Re: calculus, volume

ohh, ok.  thanks. i feel like a retard now. hehe

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#8 2006-01-13 17:00:42

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

Re: calculus, volume

No problem.  This is all fairly new to me, but hopefully someone can
show how to integrate without the table.


igloo myrtilles fourmis

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#9 2006-01-13 23:47:56

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: calculus, volume

That's just integration by recognition. We know that if you differentiated

, you'd get
.

So, if we want the answer to be half of that, we need to half the thing we're differentiating.

Using this backwards shows that

.


Why did the vector cross the road?
It wanted to be normal.

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#10 2006-01-14 03:31:40

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: calculus, volume

V = ∫2πx(f(x)-g(x))dx

2π∫x - xlnx dx = π[x²]0,e - 2π[x²lnx / 2]1,e + 2π[x²/4]1,e

Note that I had to use integration by parts for g(x).

V = πe² - πe² + 2π(e²/4 - 1/4) = π/2 (e²-1) ≈ 10.0359u²

Also note the upper and lower limits for the integrations above.  We had to exclude the values of x lower than 1 for the logarithmic part because that would have been negative and subtracted from the shaded region above the x axis.  I hope that this clears up some of the questions above.

  John, note that even though you attempted to use the disk method instead of the cylindrical shell method that the volume you tried to compute was for the region below the lnx function.  If you were to use the disk method the integral above should be subtracted from the integral of y=1 function.

Last edited by irspow (2006-01-14 03:32:36)

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