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## Topic review (newest first)

krassi_holmz
2006-01-02 19:49:47

But the AREA is true.

krassi_holmz
2006-01-02 19:48:32

Oh, yes, we want to know the volume.

John E. Franklin
2006-01-02 16:55:47

In all the light green graphs of previous posts, the point (e,0) is wrong, it is (e, -1) like in krassi's hand drawing before he raised it by 1.
Also here is some more information in a new plot showing three regions to be rotated, the hardest one being the skinny area I have labelled region C.

God
2006-01-02 11:41:46

The work seems right, so if you made a mistake, I haven't noticed it. Thanks

krassi_holmz
2006-01-02 11:29:00

Please correct me if i've done a muistake.

krassi_holmz
2006-01-02 10:18:03

Simplified area:

krassi_holmz
2006-01-02 10:16:23

Area = 5.66169988597863380413031960736...

krassi_holmz
2006-01-02 10:13:12

I'm simplifying the AREA...

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

krassi_holmz
2006-01-02 09:40:35

Ready! here is it:

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.

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
2006-01-01 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.

krassi_holmz
2006-01-01 08:36:53

Picture:

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: