in this video we're given the curve y equals e to the sine x that's the yellow curve down below and
it's cut off on the interval 0 to 2 pi and it's been revolved around the x axis to trace out a
surface of revolution and then we're asked to find the surface area of it using the so-called
ribbon method so i've shown one of these ribbons we've sliced it into a thin ribbon that's a little
area contribution to the total surface area and that ribbon is located at a position of x
and i need the area of that ribbon and certainly i need the radius of it for that
and that's just the y value corresponding to that x value and that's just e to the sine x
but then i need the thickness of the ribbon and that little tilted segment has to be represented
by a little chunk of arc length called ds so then we cut the ribbon and we unroll it and that's
what this red line is over on the right it has a thickness of ds and then the length of it is
well 2 pi times the radius it's the circumference of the original ribbon so that's 2 pi
e to the sine x that means the area contribution of this little ribbon is just the area of that
unrolled ribbon that's a 2 pi e to the sine x d s now that d s has to be expressed in terms of
x and i showed this in a previous video but a small arc length contribution
is given by the square root of 1 plus the square of the derivative of the function
times dx so we need to find the derivative of our function and then square it
and when i differentiate e to the sine x i have to use the chain rule i differentiate
with respect to sine x and i get e to the sine x because e to the x is equal to its own derivative
and then i have to multiply by the derivative of sine x which is the cosine x
and i'll go ahead and square this thing right now and this gives me cosine squared x e to the 2
sine x so when i set up my area integral that's the sum of all the contributions
to the total area so the integral of all the da's then i plug in my formula for the da's
and that's going to be a 2 pi e to the sine x times d s a little arc length increment
but that's given by this formula in terms of x so i have to take the square root of 1
plus the square of the derivative of the function so a cosine squared x e to the 2 sine x
dx and then i add up all these area contributions from the starting value of x which was 0 all the
way to the finishing value which was 2 pi so we're looking at an integral that
has no analytic solution and that's why we're asked to use a computer algebra system here to
approximate the area to the hundredths place so we bounce to our computer algebra system and i'm
using wx maxima here and just to illustrate that our integral has no analytic solution
if i try to use the symbolic integration tool that's called integrate
i plug in my function so 2 times percent pi that stands for pi in maxima times percent e that
stands for e in maxima to the sine x times this ugly square root 1 plus the square of the cosine
times e to 2 sine x and then i try to add those up as x goes from 0 to 2 pi
maxima just repeats the integral back to me because it doesn't know what to do so
there's no analytic solution to this thing so instead of using the integrate command
we switch to the quad qag commands and that's the numerical integration package for maxima
the only difference in syntax is that we have to tack on a number between 1 and 6 at the end
and that just specifies which algorithm is going to be used for the numerical integration
they're all more than precise enough for our purposes so any number between 1 and 6 will work
and i hit shift enter and i get 70.090 and so on we were asked to round to the hundreds place
so we're going to say our area is approximately 70.09 and we're done if you enjoyed this video
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