Welcome to a second video on surface area of revolution
I decide to go ahead and make a second video with rotation about the y-axis
Let's go and first review our surface area formulas
member we have a formula for rotation about the x-axis as we see here
and another one about the y-axis as we see here
remember in both cases we have an option for ds
which represents the
arc length
meaning we can integrate with respect to x
or with respect to y but of course that will affect our limits of integration
as well as how we express the radius
and that is true for both
rotation about the x-axis and rotation about the y-axis
Let's go and take a look at our example
f of x equal the cubed root of eight
on interval from zero to eight
along the x-axis
and we're gonna rotate this about the y-axis so in this direction here
and we want to find the surface area of this solid
so here's the formula for rotation about the y-axis
and for this example let's go ahead and integrate with respect to y
which means we use the formula for surface area that would be
two pi
times the integral of g of y
time the square root of one plus
g prime of y squared
respect to y
so notice we have to express our radius
and the function in terms of y
so what that means we have to take this function here y equal the cube root of x and solve it for x
Let's go ahead and do that now
if we have y equal
x to the power of one-third and we cube both sides of the equation we'll have
y cubed equals x
so using function notation this tells us that g of y
is equal to y cubed
and the derivative of g with respect to y would be equal to 3y squared
the only other information we need
is the interval for y, since were integrating with respect to y so if x goes from zero to eight
well when x is eight, the cubed root of eight would be two
the interval for y would be from zero to two
so let's go ahead and see if we can set this up
it's going to be equal to two pi
time the definite integral from zero to two of g of y which be the radius
meaning this distance here
but it has to be express in terms of y it's gonna be y cubed
times the square root of one plus
g prime of y squared
well g prime of y is 3y square so, so if we square this we'll have nine y to the fourth
respect to y
looking at this now
since the radicand is degree four
and we have a degree three term here that should key us into using u substitution
for u is going to equal one plus nine y to the fourth
so du
would be equal to
36y to the third
dy
and noticed that our integrand contains
y cubed dy
so if we want it a perfect match we should divide both sides by 36
so we have u equals one plus nine y to the fourth and 136 du
is equal to y cubed
dy
if we want to rewrite this now in terms of u we have two pi
y cubed dy will be replaced with 136 du
so we'll have the 136 here
and we have our du
and hear our radicand is equal to u
squared of u would be u to the one half
Let's go ahead and take this over the next screen and continue
Let's go ahead and integrate with respect to u
well first it should be pi over 18
then we have u to the three halves
divided by three halves
leave off the plus C because this is going to be a definite integral in terms of x
Let's go and clean this up
and rewrite it in terms of x
so here we have pi over 18 times a reciprocal of three halves that would be two thirds
well pi over 18 times two thirds
times
u to the three halves but u is one plus 9y to the fourth
need to evaluate all of this at the upper and lower limits of two and zero
Let's go ahead and clean this up one more time so it be one that would be nine
so we have pi over 27
now we'll substitute two for y
and then substitute zero for y
well two to the fourth would be sixteen times nine would be 144 plus one
145 to the three halves power
minus
over here we have
one plus zero to the three halves that would just be one
and i'll go ahead and leave it like this but let's go ahead and get a decimal approximation
and that's approximately 203.04
and this should be square units for surface area
and that's going to do it for this example I hope you found it helpful