This is a classic curve called gabriel's horn
that's a section of y equals 1 over x so technically gabriel's horn goes
all the way off to infinity but we have enough of a chunk here to look like a horn
and we're trying to do kind of everything for this so i want the arc length of this curve
and then when i actually create the solid of revolution
i want the volume of the solid and i want the surface area of that solid
so we're kind of putting all these applications together so to get the arc length um i have f of x
is 1 over x i know that i'm going to need f prime so i'm just going to compute it
right here f prime of x is negative 1
over x squared and then my arc length integral looks like
i guess i'll use this
integral as x goes from 1 to 10.
square root 1 plus this derivative squared that's 1 over x to the 4th
dx so if i go into a computer algebra system
and i'm going to go to wx maxima and i try to compute this
maxima just gives me back the same integral which means it couldn't find a way to do it symbolically
so we're going to have to use a numerical approximation on this
and that's quad qag and this function requires that you pick one of the algorithms from number
one through six that's built into maxima i usually choose algorithm number one but i'm
just going to mix it up and choose i'll go to the number two this time you're going to get the same
answer essentially to many many decimal places so i normally don't care so i get 9.15 on this
for that arc length
which is plausible because the horizontal distance from here to here from one to ten that's nine
and this is a curve that's pretty flat so i just get a tiny bit more than nine makes a lot of sense
okay let's get the volume of gabriel's horn and it looks pretty natural here to slice into discs
so there's a nice disc
it has a thickness of dx the radius of the disc so this dimension is just the distance
between the y-coordinate on the curve and the x-axis so that's just the y-coordinate 1 over x
and then i can write down a differential volume for a disk so we'll call that dv
and that's going to be the area of a face here on the disk pi times the radius squared
multiplied by the thickness of the disk and then i use an integral to add up all those dvs
dv is entirely phrased in terms of x so we're ready to go and we're integrating from one to ten
i have pi times one over x squared
dx and so i can guess the anti-derivative pretty quickly on this that's x to the
negative 2 which integrates to negative 1 over x turn it up with pi times negative
one over x evaluated from zero to ten and when i plug in the tan i get negative one tenth pi
then i subtract the result i get when i plug in oops that's not a 1.
i mean it's not a 0 it's a 1. when i plug in the 1 i end up with a positive result
and that's just this term becomes one and i just get pi so that looks like nine tenths pi
let's get the surface area of this gabriel's horn
so i'm going to mark out a little ribbon that i want to cut out of the surface
so it's important when we're doing surface area
to use ds a little section of arc
for the thickness instead of dx
and i'm going to slice and unroll this thing maybe before i do that i'll just write down the radius
of the ribbon it's the same as the previous part the distance here is just one over x
so i'm going to slice and unroll this little ribbon
and lay it out flat and the length of that is going to be the circumference
of the rhythm that it came from so that's 2 pi times the radius
or 2 pi times 1 over x the thickness is a little d s a little arc element
and remember that comes out to this expression that we used to add up all the arc a minute ago
one over one plus one plus one over x to the fourth
and now i can write down this infinitesimal area contribution d a is going to be
two pi times one over x square root one plus one over x to the fourth
dx so now i can use integration to add up all those d
a's
okay and interestingly enough it turns out this integral has a symbolic solution um i haven't even
attempted it by hand but let's go over to maxima and check out what it looks like so i plugged
in and i said integrate this thing because i was curious if if it had a symbolic solution and press
shift enter and i get that exact symbolic solution looks like it must have been a pretty hard
integral i could get a decimal approximation for this just by using the float command or i could
go back to the beginning and change to quad qag to just use a numerical integration technique but
since it came out in exact form i can go ahead and use float so 15.2 ish i'd say is good enough
you