Math Is Fun Forum / Help with 3D rotation

God · 2005-12-31 10:33:51

Given the function:
ƒ(x) = e^x-ln(x)-x^x on the interval 1/e <= x <= e

I now have to rotate this around the line x + y = 1

What will be the volume of the solid?

Just for a visual:

Thanks

Last edited by God (2005-12-31 11:12:32)

krassi_holmz · 2005-12-31 10:52:29

Plot:

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Last edited by krassi_holmz (2005-12-31 11:44:25)

God · 2005-12-31 11:17:05

? I don't get it

Last edited by God (2005-12-31 11:17:18)

krassi_holmz · 2005-12-31 11:17:07

Is this write?
I just orbitained it:

God · 2005-12-31 11:21:44

See the problem, to be more specific, is that I can't figure out how to integrate ƒ(x)... esp. the x^x

krassi_holmz · 2005-12-31 11:28:01

You can't integrate it exactly.

krassi_holmz · 2005-12-31 11:30:53

It's just coordinate change.

krassi_holmz · 2005-12-31 11:41:05

We must find the equation of the line in the coordinate system, roated 45 degrees ++clock:

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Last edited by krassi_holmz (2005-12-31 11:45:16)

John E. Franklin · 2005-12-31 11:44:14

I think you could divide the area up into parts as a start.

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krassi_holmz · 2005-12-31 12:00:42

If we use polar coordinates is easy:
Let A= (r, alfa)pol
Then
A'= (r,alfa+45 deg)pol= (r,alfa+(1/8)2Pi rad)pol= (r,alfa+Pi/4)pol.
Now from decart c.s. to polar c.s.:
Let A= (x,y)dec
r=sqrt(x^2+y^2)
cos(alfa)=y/r
alfa=arccos(y/r)=arccos(y/sqrt(x^2+y^2))

Then
A'= (sqrt(x^2+y^2),arccos(y/sqrt(x^2+y^2))+Pi/4)pol
Now from polar to decart:

x'=r'*sin(alfa')=sqrt(x^2+y^2)*sin(arccos(y/sqrt(x^2+y^2))+Pi/4)
y'=r'*cos(alfa')=sqrt(x^2+y^2)*cos(arccos(y/sqrt(x^2+y^2))+Pi/4)

Last edited by krassi_holmz (2005-12-31 12:07:35)

krassi_holmz · 2005-12-31 20:35:23

Here is it:

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Last edited by krassi_holmz (2005-12-31 20:36:47)

krassi_holmz · 2005-12-31 20:37:25

Symplifyng...

krassi_holmz · 2006-01-01 01:06:38

Done:

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Last edited by krassi_holmz (2006-01-01 01:10:01)

John E. Franklin · 2006-01-01 02:21:30

I don't understand a lot of this rotation, but here are some questions.

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John E. Franklin · 2006-01-01 02:32:07

Also I was wondering if you considered that if the function is rotated,
some of the new vertical values are directly above one another, thus
it is not a one-to-one function.

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Last edited by John E. Franklin (2006-01-02 07:52:02)

krassi_holmz · 2006-01-01 02:50:02

For the first -you're right.
For the second-no.
I'm starting explaining everything.

krassi_holmz · 2006-01-01 04:20:52

1. Coordinate transforms:
Let the point A has coordinates {a,b} in coordinate system xOy.
We must find the coordinates of a in coordinate system x'Oy', which is xOy rotated 45° ++clock:

krassi_holmz · 2006-01-01 08:36:53

Picture:

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Last edited by krassi_holmz (2006-01-01 08:38:03)

krassi_holmz · 2006-01-01 08:42:44

As you see the coordinates of A in the second coordinate system A=={x',y'}II are equal to the coordinates of point A1 =={x',y'}I, which is A, rotated 45deg --clock.
So we must find what will be the coorinates of point A{x,y}, when we rotate it 45deg --clock.

John E. Franklin · 2006-01-02 07:46:19

slope = y' = e^x - 1/x - x^x(1+lnx)

By looking at a graph, it looks like x=.67 is about horizontal slope, so
try .67 for x to see if is nearly zero slope., and I get slope = 0.003267, which is awesome!!

Now try 1/e to see if it is slope of -1.
try x = .367879441, and I got slope is -1.2736, thus the tangent with slope -1 is to the right a little bit.
What is this number for x??
Will try to find this x with a BASIC program. Got x = 0.4112922
From this we can conclude that "area 4" is non-zero.
So we have 4 regions to rotate around y=-x+1 to find the volume.

krassi_holmz · 2006-01-02 08:55:25

Ha, ha, ha!
I tougth a while and I got very simple geometric solution.
But you must wait a wnile to make a pictures.

krassi_holmz · 2006-01-02 09:40:35

Ready! here is it:

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Last edited by krassi_holmz (2006-01-02 09:41:30)

krassi_holmz · 2006-01-02 09:49:38

And here's the proof thingy:
Let ff[x]=ƒ(x)
The wanted area won't change if we push it one up. We to this to reduce ff to positive function.
We want S+S1+S3.
(x,y) means point x,y.

But
(rectangular with 45 deg)

so S = INEGRAL - S2.

Now we'll find S1 and S3:
(rectangular with 45 deg)
(rectanguler with 45 deg)

Then

Last edited by krassi_holmz (2006-01-02 10:05:20)

krassi_holmz · 2006-01-02 10:13:12

I'm simplifying the AREA...

krassi_holmz · 2006-01-02 10:16:23

Area = 5.66169988597863380413031960736...

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#1 2005-12-31 10:33:51

Help with 3D rotation

#2 2005-12-31 10:52:29

Re: Help with 3D rotation

#3 2005-12-31 11:17:05

Re: Help with 3D rotation

#4 2005-12-31 11:17:07

Re: Help with 3D rotation

#5 2005-12-31 11:21:44

Re: Help with 3D rotation

#6 2005-12-31 11:28:01

Re: Help with 3D rotation

#7 2005-12-31 11:30:53

Re: Help with 3D rotation

#8 2005-12-31 11:41:05

Re: Help with 3D rotation

#9 2005-12-31 11:44:14

Re: Help with 3D rotation

#10 2005-12-31 12:00:42

Re: Help with 3D rotation

#11 2005-12-31 20:35:23

Re: Help with 3D rotation

#12 2005-12-31 20:37:25

Re: Help with 3D rotation

#13 2006-01-01 01:06:38

Re: Help with 3D rotation

#14 2006-01-01 02:21:30

Re: Help with 3D rotation

#15 2006-01-01 02:32:07

Re: Help with 3D rotation

#16 2006-01-01 02:50:02

Re: Help with 3D rotation

#17 2006-01-01 04:20:52

Re: Help with 3D rotation

#18 2006-01-01 08:36:53

Re: Help with 3D rotation

#19 2006-01-01 08:42:44

Re: Help with 3D rotation

#20 2006-01-02 07:46:19

Re: Help with 3D rotation

#21 2006-01-02 08:55:25

Re: Help with 3D rotation

#22 2006-01-02 09:40:35

Re: Help with 3D rotation

#23 2006-01-02 09:49:38

Re: Help with 3D rotation

#24 2006-01-02 10:13:12

Re: Help with 3D rotation

#25 2006-01-02 10:16:23

Re: Help with 3D rotation

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