All right let's find out how to get the surface area of a surface of revolution so what we've
done here is we've taken a little section of the function f of x and then we revolved it around
the y axis we spun it around this way and traced out this surface so the surface has axial symmetry
and we want to get the area of that weird curvy surface
so we take a physical integration kind of approach to this
and we're going to slice it into infinitesimal little elements that hopefully we can
frame in terms of a single integration variable
and then we should be able to use an integral to add them all up
all right so to be clear i've got to draw the back half of this because
this is not a disc it's a ribbon because this is a surface and not a solid
now there's a little thickness to the ribbon that i showed there and
if i pop this out i just want to show it a little bit bigger
it'll go like that
there's that little thickness to the ribbon and it is superimposed on the curve of f so
f continues out of this side and continues out of this side
that little ribbon or a little ribbon thickness we're going to call that ds
and we're going to break that into x and y components this is very similar to the derivation
of the arc length formula and the x part will be called dx and the y part will be called d y
so now i can find a relationship between d s d x and d y it's just the pythagorean
theorem so i'm going to write d s as the square root dx squared plus d y squared
and then depending on the formula for the curve i'm i'm given
in this case i have an f of x and so what i want to do is divide
divide out a dx squared or factor it out there's another way to say it
and the reason i want to do it that way is because it leaves me with a dydx
in there and that's just f prime of x before i get ahead of myself there's my dx squared
when i factor out the square root of dx squared that's just dx
so i have the square root 1 plus f prime of x
squared dx and so that little contribution
to arc length or the thickness of this ribbon is given by that expression
all right so now we're going to cut and unroll the ribbon so i'm going to slice it right here
and we'll cut and unroll when i do that i get a skinny rectangle
and a little thickness we just found is right there that's ds
then if i go back to my diagram there's a bit more to do this is just some
location x some horizontal coordinate
and so that distance from the y axis to that location x
is actually the radius of the little ribbon so the ribbon has a length of 2 pi times the radius
so that'll give me that's 2 pi x right here
and i like to call this little area contribution d a and that's going to be 2 pi x d s so we're
getting close now um i need to go ahead and and put in i'm changing colors too much here i go
i need to go ahead and put in the definition of ds in terms of x over here so i get 2 pi
x square root 1 plus f prime of x squared dx now that that's all phrased in terms
of a single variable it's ready for the integration so finally i'll write that a
is equal to the sum of all the contributions to a so again integration is a is a summation device
and it just adds up all these infinitesimal contributions to the total area
um and i can put in my little d a my formula from d a 2 pi x root
1 plus f prime of x squared dx as x goes from whatever the limits are i'll just say x1 to x2