Surface Area (Surface of Revolution) - Discrepancy

[DoubleMike]

So this question has been bothering me for a very long time... but only recently have I mustered up courage to register to ask it.

Before though, let me draw a parallel analogy, so you can see where I'm coming from. When you take the volume of a region enclosed by one or more functions revolved about a line, either the method of diks or shells is used.

Let us examine the volume of a sphere. The simplest way to calculate the volume of a sphere of radius R is to use the disk method with limits of integration -R to R, in respect to either variable (due to the symmetry).

So in essence the Riemann sum is the addition of the volume of infinitely many cylinders. These have parallel sides to the axis of the sphere/integration. Though strictly speaking we are adding very small volumes, I'd rather think of it as adding an infinite number of areas. This goes well with my understanding of infinities (as I've entreated in the General Math Forum). If this is flawed however, please feel free to elucidate.

Anyway... When asked to take the surface area of a sphere, my book goes about it a very strange method indeed. It breaks the sphere up into an infinite number of partial cones, and uses the formula of their lateral surface area. I understand why this works... it makes sense. But why won't this work:

Breaking the sphere up into an infinite number of cylinders, and using the surface area formula for those (translational surface of a circle).

Here I have supplied a picture of what I mean:

http://www.geocities.com/boeclan/sphere.jpg
*(if the link doesn't work, copy and paste into address bar)

That picture goes for both the surface area and volume integrals. In the case of volume, that is the correct cross-section, whereas for surface area it is not.

Why is it that the function's slope is ignored when calculating volume, but for surface area, one must use the circuitous method described by my book?

*edit = typo

 

[HallsofIvy]

So what you are saying is that you are dividing the sphere into very thin disks, of height dh and radius "r" where, since the sphere is given by x²+ y²+ z²= R², r= $\sqrt{R^2- y^2}$. The lateral area of such a cylinder is given by [itex]2\pir dy= 2\pi\sqrt{R^2-y^2}dy.

Integrate $2\pi\int_{-R}^R\sqrt{R^2-y^2}dy$ and see what you get.
(If you are right, it will be 4πr².)

 

[DoubleMike]

Is the integral $2\pi\int_{-R}^R\sqrt{R^2-x^2}dx$?

How would go integrate that without resorting to trigonometric substitution?
Well anyway, I checked the above integral for a random radius and it differs from the actual surface area!

 

[dextercioby]

Is the integral $2\pi\int_{-R}^R\sqrt{R^2-x^2}dx$?

How would go integrate that without resorting to trigonometric substitution?
Well anyway, I checked the above integral for a random radius and it differs from the actual surface area!

Write it (leave the 2pi for the moment):
$$\int_{-R}^{+R} \frac{R^{2}-x^{2}}{\sqrt{R^{2}-x^{2}}} dx$$

and then use the fact that
$$\int \frac{du}{\sqrt{1-u^{2}}}=\arcsin u +C$$

and partial integration to get the result...

Daniel.

P.S.Use the fact that the integrand is even and change the lower integration limit correspondingly.

 

[mathwonk]

a smarter way to do the surface area of a sphere is by analogy with the method you used for the volume, i.e. by finding the derivative of the volume formula.

i.e. if one grows the volume of a sphere outwards it follows that the derivative of the volume function is the area of a cylindrical slice. Thus the volume itself is the integral of the area formula for this cylindrical slice, the so called "cylindrical shells" method.

On the other hand if one grows the volume of a sphere radially, it follows that the derivative of the volume function, wrt the radius is the area function of the sphere. hence the derivative of volume wrt radius is area for a sphere.

now this does not help compute the volume since we are not given the area, but since we already know the volume it does let us compute area, i.e. the area formula for a sphere, is the derivative of (4/3)pi r^3, wrt r, namely 4pi r^2.


you might like to read up on pappus theorem as well, for another analog of computing area via slices.

 

[DoubleMike]

I'm not entirely sure I know what you mean by even integrand. Also, I think trigonometric substitution is simpler in this case, but I keep messing it up:

$2\pi\int_{-R}^R\sqrt{R^2-x^2}dx$
I constructed a triangle with hypothenuse R and legs x and $\sqrt{R^2-x^2}$

so $\sin\theta = \frac{\sqrt{R^2-x^2}}{R}$ and $\cos\theta = \frac{x}{R}$

hence $-R\sin\theta d\theta = dx$ and $R\sin\theta = \sqrt{R^2-x^2}$

after the appropriate substitutions I get

$2\pi R^2 \int_{0}^\pi \sin^2 \theta d\theta$
using a power reducing formula
$\pi R^2 \int_{0}^\pi 1+\cos 2\theta$
$\pi R^2 [\theta + \frac{1}{2}\sin 2\theta]_{0}^\pi$
$\pi R^2 [\pi + 0]$
$\pi^2 R^2$

I'm not entirely sure of my math there, I didn't get the correct answer before, and working it out on itex was even worse... I might very well be wrong.

Regardless, integrating on my calculator, the method of disks differs largely from the actual formula derived using the slope and lateral surface of cones. What gives?

 

[Justin Lazear]

You're integrating just fine. Your cylinders model is incorrect, though.

Your model doesn't account for the fact that the length element ds of the cylinder must be longer, since it's at an angle. Compare the arclength traced out by the circle in element dx at x = 0 and x = near R. The first is just about straight, but the second is at a large angle, so it also goes down a good bit in the length dx, and therefore it should be longer.

--J

 

[DoubleMike]

Yeah I realize that, but why do you have to account for the angle when finding the surface area, whereas for volume you can pretty much ignore it?

I don't understand that, besides... As the differential of x approaches 0, the significance of slope should be negligible.

 

[Justin Lazear]

You do have to account for it. That's why when you change to cylindrical coordinates you get an $r d\theta$ term or an $r^2 \sin{\theta} dr d\theta$ term in spherical coordinates.

--J

 

[mathwonk]

i am trying to actually answer the question of how to view the calculation of surface area as analogous to that of volume, but i do not seem to be making any impression on the questioner. oh well. cest la vie. people would rather calculate than think.

 

[Justin Lazear]

i am trying to actually answer the question of how to view the calculation of surface area as analogous to that of volume, but i do not seem to be making any impression on the questioner. oh well. cest la vie. people would rather calculate than think.

Thinking is hard. Calculating is easy. Or at least that's what's taught in high school.

--J

 

[DoubleMike]

I'm not sure whether to be insulted or not... Perhaps a better explanation is in order...

As it stands, I have to yet see a reason why I have to account for the curvature (besides the fact that the numbers don't match up). Yeah I understand that the curvature becomes more and more severe... But this never seemed to be a problem for calculating volume... And even if it is, why is it that $\pi \int_{-R}^R R^2-X^2 dx$ is a valid integral for volume? I don't see any reference to the slope of line tangent to the sphere's surface...

 

[mathwonk]

Basic suggestion: when you ask a question, and someone answers it for you, it may be wise to ask for more details when the answer goes over your head.

 

[DoubleMike]

Ok, slow down.. Let's make up. I'm sorry I didn't read your post as carefully as I should have.

I understand what you're saying though, I had thought about it before (I fancied it suspicious that the derivative of volume happened to be the surface area.) I haven't done anything with spherical coordinates yet, but I follow the reasoning.

But still, that doesn't explain why breaking the sphere into small cylindrical segments and taking their lateral surface area doesn't work.

I know that the curvature is important, but there is nothing in the volume's Riemann sum that refers to it... So why should surface area be any different?

 

[Justin Lazear]

I'm not sure whether to be insulted or not... Perhaps a better explanation is in order...

As it stands, I have to yet see a reason why I have to account for the curvature (besides the fact that the numbers don't match up). Yeah I understand that the curvature becomes more and more severe... But this never seemed to be a problem for calculating volume... And even if it is, why is it that $\pi \int_{-R}^R R^2-X^2 dx$ is a valid integral for volume? I don't see any reference to the slope of line tangent to the sphere's surface...

The slope of the lines in the Cartesian coordinate system is always 1! dxdydz is always just a big box! As I said before, the case is not the same for coordinate systems where your dV element is not always just a big (edit: little) box, and in those coordinate systems, you do have to account for curvature.

--J

 

[mathwonk]

what i am saying is there is no reason why what you suggested should work, since it is not really analogous to the method of calculating volume. in my post i give two methods which ARE analogous. i.e. if one works so should the other.

in your post, you give two rather different methods and ask why one works and the other does not. i ask you to think about why the area computation you suggest should work. i.e. what does it have in common with the shells method of computing volume?

and i apologize for getting my nose in a snit. i tried to edit but too slowly.

ill be back later but right now duke / wake is on.

 

[DoubleMike]

Ok, so perhaps a better question to ask it why the volume of sphere doesn't require that argument.

as it stands the Riemann sum is
$\pi\sum (R^2 - x^2) \delta x$ (not sure how to make the triangle on itex)

why not a sum of some function that takes the arguments radius and slope to calculate the volume of infinitely small cone sections?

$\pi\sum f(x, \frac{dy}{dx}) \delta x$

though it came to my mind that the function for the volume of a cone was derived using disks, so it too would be compromised.

Anyway, the point being that for a sphere, as you sum up the disks, shouldn't the fact that they have parallel sides skew the integral? (Because they don't follow the sphere's curvature.)

I don't know how to express my question!

 

[mathwonk]

what i mean is that calculating voluimes has nothing to do with approximations by cylinders or disks, it has to do with defining a growing volume function along some axis or other, and finding the derivative. if you consider a family of discs with enlarging radii, centered along the z axis, and let V(r) be the volume of your figure intersected with the solid cylinder of radius r, then the derivative is the area of the intersection of your figure intersected with the shell of radius r. this is called the method of cylindrical shells.


On the other hand if you consider a growing cylinder of fixed radius and growing height, intersected with your figure, then the derivative wrt height is the area of your figure intersected with the top circle of the cylinder of than height. then it is called the method of discs.

If you consider a famliy of solid spheres growing radially outward, intersected with your figure, then the derivative is the area of you figure with the surface area of the sphere of that radius. this method is not usually taught hence has no name.

But basically these methods for finding volumes have nothing to do with the shape of the family of figures that are growing, only with the relationship between their volume and the area of their leading face. I.e. these methods of calculating volumes are based on the fact that the derivative of a growing family of volumes is the surface area of the leading face of the family of figures. (provided the leading face is perpendicular to the direction of growth.)

In Pappus method for instance, we grow the volume in a circle by revolving a disk about the z axis, and consider the intersection of our figure with the solid partial torus generated by revolving the disc through a given angle. the derivative is the intersection of our figure with the area of the leading face, i.e. the moving circle.


Thus this method is much smarter for computing the volume of a torus generated by revolving a circle, since the derivative, the area of the moving circle is constant. thus the volume is that area times the circumference of the circle of centers of the moving circles.

so the whole point is to come up with a moving volume function, actually compute its derivative, usually an area, and then integrate back to get the volume.

the fact that the usual moving volume functions have derivative either the area of a cyliunder or a disc, is due to lack of imagination of textbook writers.


In particular, it is not a consequence of that accident that one should do all calculations by approximating things with discs and cylinders.

rather one should try to compute ones desired quantity by some kind of appropriate moving function to the given problem, and then try to figure out what its derivative is.


so for example to compute the area of a torus, generate by revolving a circle, the derivative is delta area/ delta circumference of path of revolution, = circumference of the circle being revolved, again a constant. so the area of a torus equals the length of the circle being revoolved times the circumference of the path of revolution.

this is pappus method from thousands of years before calculus, explained by virtue of Newton's ideas from calculus.

these things are not explained well in textbooks because most textbook writers just copy what other writers have said, without thinking about it first.

the methjod of computing areas using tangential cones is based on another entirely different idea they do not choose to explain. the key point is that since nbow the approximating regions no longer lie inside the given region, they mkust at elast be tangent to it.


tyhere is abetter way to view it absed on aparmetrizations, and change of area formulas under paramewtrizations, but enoiugh for now.

your question by the way is very intelligent and shows wonderful curiosity.

 

[mathwonk]

so ultimately the point is to understand why things are true. as presented in some books, the volume calculation is just, well these cylinders approximate that volume so in the limit we get the actual volume. oh yeah? why? what's the proof?

if you actually understand the proof of why the cylinder calculation does give the volume, then try to make that same proof go through for area, you should see that something crucial is then missing in the second case.

of coure it helps if you have a definition of volume and area.

but the method i am giving comes equipped with an argument. i.e. define a volume function V(r) or V(z) or V(something), and then compoute dV/dsomething as an area. then integrate back to get volume.

in the books they are skipping the explanation of why the area of that cylinder or of that disc, is really dV/dr or dV/dz, for some volujme function.

But if it were not, then you not get volume by integrating it.

so in the case of surface area, you are trying to compute surface area as a limit of areas of a different type.

what if you approximated your surface by discs that stuck straight out of your surface? would the areas of those discs have any relation to the area of your surface?

the better analogy for your problem, is why does the computation of arclength work?

i.e. the computation of arclength does work by approximating the curve by straight lines.

now if you look under the radical of the integrand for arclength you will see that what is being integrated is the length of approximating tangent line segments to your curve.

this is the analogy that does generalize to surface area. i.e. if you approximate your aurface by tangential pieces of surface, then their areas will approximate the surface area.

best wishes.

sorry if these explanations are inadequate, but they are the best i have at the moment.

 

[mathwonk]

actually the arclength anal;ogy completely explains the surface area method, since if a piece of tangential segment is a good approximation to the curve then the cone obtained by revolving it is a good approximation to the surface area of revolution.

i.e. if the arclength is the integral of ds then the surface area is the integral of 2piRds.

 

[DoubleMike]

Yeah I'm still trying to understand you completely... I'll best tackle it tomorrow. For now though, before I do what you suggested (trying to prove the sum for the surface area) I'd like to ask, what makes disk slices a good enough approximation to warrant being integrated for volume?

The way my book explained it (or the way I understood it rather), is that you could slice anything whichever way you wanted and integrate the sum of those slices as their number approached infinity. Maybe that's what's wrong with my understanding.

 

[Justin Lazear]

Intuitively, the error drops to zero when integrating volume with the discs, since the error depends on the height of the disc. Since we're letting this height become arbitrarily small, i.e. it becomes $\Delta x$ (use \Delta for the triangle, not \delta) the error is directly proportional to $\Delta x$, and so is therefore also arbitrarily small. If we take the limit as this height goes to zero, the error goes to zero.

To see what I mean, consider the volume of the disc element and the actual volume between x and x + $\Delta x$. For a sufficiently small choice of $\Delta x$, the slope of surface of the volume is pretty much constant, so we can consider it to be so. Therefore, the actual volume element is a frustum (http://mathworld.wolfram.com/ConicalFrustum.html). The disc volume element is just a plain cylinder, and the difference between the two is that extra little triangle at the edge rotated about the axis. The aforementioned Pappus Theorem tells you that this difference in volume is equal to $2\pi r A$, where A is the area of that triangle, which is just $\frac{1}{2}\Delta x (f(x + \Delta x) - f(x))$, which is proportional to $\Delta x$. It's clear that $E = 2 \pi r \frac{1}{2}\Delta x (f(x + \Delta x) - f(x)) \to 0$ as $\Delta x \to 0$. So the integral of the cylindrical dV element will give us the true area.

The same argument does not apply for the surface area, though. The error term is constant relative to $\Delta x$, so we must include it, since it does not drop out as $\Delta x \to 0$. You can show that one to yourself.

--J

 

[mathwonk]

the simplest case is the proof of the FTC for why area is the integral of height.

assume for simplicity the height is increasing.

then a small change in x produces a change in area under the curve, that is caught between two rectangles, both with base deltax, and one with height f(x) and the other with height f(x+deltax). so deltaA, the change in area is caught between
deltax times f(x+deltax), and deltax times f(x). now to compute dA/dx, we divide delta A by deltax, and then take the limit as delta x goes to zero.

but since deltaA is caught between deltax times f(x+deltax), and deltax times f(x), it follows that deltaA/deltax is caught between f(x) and f(x + deltax). now both of these converge to f(x) as deltax goes to zero,because f is assumked continuous.

so by the squeeze law, also deltaA/deltax also converges to f(x).



In the same way deltaV, change in volume is caught between (cylindrical shells method) deltax times (2pi[x+ (1/2)deltax]) times f(x),
and deltax times (2pi[x+ (1/2)deltax]) times f(x+deltax)


thus the difference quotient deltaV/deltax is caught between
(2pi[x+ (1/2)deltax]) times f(x),
and (2pi[x+ (1/2)deltax]) times f(x+deltax).

now as deltax goes to zero, both of these converge to 2pi xf(x), again because f is continuous.

so to get the volume, you integrate back 2pi xf(x). (the area of a cylinder).


now your attempt at calculating the surface area uses no such "over and under" approximation of the area. i.e. I am guessing your book did a cruddy job of explaining why the cylinder method works, it just said, "well this approximation is close enough", but never said why it works.


this kind of thing bothered me for years teaching the course until i finally grokked it for myself. you will not find the antiderivative ideas i am discussing here in many books. i first got on this wavelength from reading the fine book by joseph kitchen, once one of the very best teachers at harvard and then at duke in the 1960's. his book was "different", hence not widely used i guess, and now almost unavailable. i could not find a copy tonight while searching for various used books to recommend to people here.

 

[mathwonk]

recalling Professor kitchen again, his teaching evaluations were about the best I have ever heard of. One sentence I recall from an independent summary of them: " A large minority of Professor Kitchen's students think that he is God."