Here is the computation I wanted to show at the end of today’s lecture: Consider a function that is continuous and invertible on an interval . Rotate about the -axis the region bounded by , , and . Then the values assigned to the volume of the resulting solid by applying either the disk method or the shell method coincide.

To see this, let’s call the expression obtained using the disk (or washer) method and the expression obtained using the shell method.

Then and .

As I mentioned in lecture, the idea to show the equality of both expressions is to, more generally, consider functions and that compute the values given by the disk and shell methods, respectively, for the volume of the solid obtained by rotating about the -axis the region bounded by , , and . Here, is a new variable that varies in . Hence, and . We will show that for all .

The trick is to proceed indirectly and, rather than looking at and , we examine their derivatives. Of course, if for all , then we must also have . Let’s check that this is indeed the case:

We have . To compute , let’s simplify a bit before differentiating: , so (Notice that does not depend on , so we can take out of the integral sign.)

Then so , or , since .

We have found that .

This does not automatically imply that , but almost: Let . Then (for all ), so is constant.

Finally, notice that . Since is constant, it follows that for all . And we are done.

Remark 1. We will see a different argument later in the course, once we study integration by parts.

Remark 2. One can also check that the disk and shell methods provide the same result when we rotate about the -axis the region bounded by , and . To check your understanding of the argument above, it may be useful to try to work this case out on your own; the algebra is somewhat simpler than the computations I just detailed.

This entry was posted on Thursday, August 28th, 2008 at 3:50 am and is filed under 175: Calculus II. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]

A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

Sure. A large class of examples comes from the partition calculus. A simple result of the kind I have in mind is the following: Any infinite graph contains either a copy of the complete graph on countably many vertices or of the independent graph on countably many vertices. However, if we want to find an uncountable complete or independent graph, it is not e […]

I think that, from a modern point of view, there is a misunderstanding in the position that you suggest in your question. Really, "set theory" should be understood as an umbrella term that covers a whole hierarchy of ZFC-related theories. Perhaps one of the most significant advances in foundations is the identification of the consistency strength h […]

I'll only discuss the first question. As pointed out by Asaf, the argument is not correct, but something interesting can be said anyway. There are a couple of issues. A key problem is with the idea of an "explicitly constructed" set. Indeed, for instance, there are explicitly constructed sets of reals that are uncountable and of size continuum […]

The question seems to be: Assume that there is a Vitali set $V$. Is there an explicit bijection between $V$ and $\mathbb R$? The answer is yes, by an application of the Cantor-Schröder-Bernstein theorem: there is an explicit injection from $\mathbb R$ into $\mathbb R/\mathbb Q$ (provably in ZF, this requires some thought, or see the answers to this question) […]

If a set $X$ is well-founded (essentially, if it contains no infinite $\in$-descending chains), then indeed $\emptyset$ belongs to its transitive closure, that is, either $X=\emptyset$ or $\emptyset\in\bigcup X$ or $\emptyset\in\bigcup\bigcup X$ or... However, this does not mean that there is some $n$ such that the result of iterating the union operation $n$ […]