Here is an animation which may help you imaging a shape which has a circular base, with parallel slices perpendicular to the base being equilateral triangles:
The same thing, where the slices are squares.
And here is the region in the (x,y) plane between 
I have seen some people try to blindly use the formulae for volumes of revolution by cylindrical cross-sections and by cylindrical shells, and I thought that I would write a guide as to how I would recommend tackling such problems, as generally just using the formulae will lead you down blind alleys.
I’ve created an example, with an animation, which I hope will help to master this technique.
So, here is a relatively fool-proof strategy:
In the most recent tutorial there is a question about converting a Riemann sum to a definite integral, and it seems to be tripping up quite a few students. I wanted to run through one of the calculations in detail so you can see how to answer such a question.
Let’s look at the example:
There are many ways to tackle such a question but let’s take one particular path. Let’s start by the fact that when the limit is defined, the limit of a sum is the sum of the limits. We can split up our expression into 3, which looks like:
Let’s tackle each of these separately. Let’s look at the first term:
Well, we can take the factor of 13 outside the front of the whole thing to start with, along with the factor of 9, and this will give
We see here that we have a sum of terms, and a factor which looks like 
From http://press.princeton.edu/titles/11024.html
This book was sent to me by the publisher as a review copy.
PHILOSOPHY OF MATHEMATICS OR PHILOSOPHY FOR MATHEMATICS? By Henri Laurie.
Review of Øystein Linnebo’s “Philosophy of Mathematics”, Princeton University Press, 2017. (This one is impressionistic; I hope to present a more conventional summary-of-contents review in due course).
I’ve just read Øystein Linnebo’s superb book on the philosophy of mathematics. It is very, very good. Superbly clear, concise, well organised, it gives not only a very accessible introduction but also takes the reader all the way to the cutting edge of what philosophers are doing in the philosophy of mathematics. Above all, Linnebo writes as a fully engaged philosopher and makes his preferred choice of philosophical position clear. But this is no mere polemic: I felt he clearly and forcefully presents the strengths and weaknesses of all the philosophical positions he discusses.
That said, even an introductory text in philosophy these days is not always easy reading.…
During tutorials last week, a number of students asked how to understand identities that are used in the calculation of various Riemann sums and their limits.
These identities are:
Let’s go through these one by one. We must first remember what the sigma notation means. If we have:
It means the sum of terms of the forms f(i) for i starting with 1 and going up to i=n. Sometimes n will actually be an integer, and sometimes it will be left arbitrary. So, the above sum can be written as:
We haven’t specified what f is, but that’s because this statement is general and applies for any time of function of i. In the first of the identities above, the function is simply f(i)=1, which isn’t a very interesting function, but it still is one. It says, whatever i we put in, output 1. So this sum can be written as:
Where there are n terms.…
The distance problem
If I want to know how far I walked during an hour, I can ask how far I walked in the first five minutes, and how far I walked in the second five minutes, and how far I walked in the third five minutes, etc. and add them all together. ie. I could write:
Where 
where you can think of velocity as the same thing as speed (though there are subtle differences which you will find out about later). This formula works if the velocity is constant, but what if it is changing. Well, if we have a graph of velocity against time, then we can think about splitting the graph into intervals (like the five minute intervals above), and approximating that during a small interval of time, the velocity is roughly constant.…
I wanted to put up a little summary of some of the most important things to remember from the end of last semester. There was a sudden input of new concepts, so let’s put some of them down here to get a clear reminder of what we need to know. A few things in this post:
Antiderivatives
An antiderivative of a function 
Note that we say an antiderivative, not the antiderivative. There can be many functions whose derivatives give the same thing. While we know that:
and therefore 
So 
This book was sent to me by the publisher as a review copy.
This is a book of some impressive magnitude, both in terms of the time span that it covers (being millennia), as well as the ways in which it discusses the context and content of the ciphers, most of which, as the title suggests, are unsolved. The book starts with perhaps the most mysterious of all unbroken ciphers: The Voynich Manuscript (the entirety of which can be found here). This story in itself is perhaps the most fascinating in the history of all encrypted documents, and that we still don’t know if it truly contains anything of interest, or is just a cleverly constructed (though several hundred year old) hoax makes it all the more intriguing.
The writing rather effortlessly weaves between the potential origin stories, the history of the ownership of the manuscript and the attempts to decode it.…
I have had a lot of conversations with students over the last couple of weeks which made me want to write this post. I apologise in advance that it will be rather long.
It’s also important to state that this message doesn’t hold for everyone, but it is worth seriously thinking about.
Many students recently have asked me about how to go through tutorials, and how to revise. They are finding that while they are sitting down for a long time with their tutorials, the tests still feel really hard.
To an important extent, technologies have changed the way we think and act over the last couple of decades. In many ways, things were easier in my day when we didn’t have so many technological distractions. Cellphones were rare when I was a student and smartphones were still over a decade away! There was no Facebook, or Youtube, or Instagram.…
The Department of Maths and Applied Maths at UCT has a position open at the level of lecturer or senior lecturer. The advert can be found here:
Applications in all areas of Mathematics and Applied Mathematics will be considered.
Minimum requirements include:
For the level of Lecturer:
For the level of Senior Lecturer:
Responsibilities include:
The annual remuneration package for 2017, including benefits are:
To apply, please e-mail the documents listed below in a single pdf file to Ms Vathiswa Mbangi at recruitment04@uct.ac.za
An application which does not comply with the above requirements will be regarded as incomplete.
