Introduction to trigonometric substitution

I have decided to start writing some posts here, and this is my first post. I would like to introduce trig substitution by presenting an example that you have seen before. Trig substitution is one of the techniques of integration, it’s like u substitution, except that you use a trig function only.

Let’s get into the example already!

\int_{-1}^{1} \sqrt{1-x^2} dx

If you equate the integrand to y (and get x^2+y^2=1, y\geq 0), you should be able to see that this is the area of the upper half of a unit circle. The answer to this definite integral is therefore the area of the upper half of the unit circle (yes, the definite integral of f(x) from a to b gives you the net area between f(x) and the x-axis from x=a to x=b), is \frac{\pi}{2}.

We relied on the geometrical interpretation of the integral to solve the definite integral, but can we also show this algebraically?…

By | August 27th, 2017|English, First year, MAM1000|3 Comments

Riemann sums to definite integral conversion

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:

 

\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(9\left(4+(i-1)\frac{6}{n}\right)^2-8\left(4+(i-1)\frac{6}{n}\right)+7\right)\frac{13}{n}

 

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:

 

\lim_{n\rightarrow\infty}\sum_{i=1}^n9\left(4+(i-1)\frac{6}{n}\right)^2\frac{13}{n}-\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(8\left(4+(i-1)\frac{6}{n}\right)\right)\frac{13}{n}+\lim_{n\rightarrow\infty}\sum_{i=1}^n7\frac{13}{n}

 

Let’s tackle each of these separately. Let’s look at the first term:

 

\lim_{n\rightarrow\infty}\sum_{i=1}^n9\left(4+(i-1)\frac{6}{n}\right)^2\frac{13}{n}

 

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

 

13\times 9\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(4+(i-1)\frac{6}{n}\right)^2\frac{1}{n}

 

We see here that we have a sum of terms, and a factor which looks like \frac{1}{n} in each term.…

By | August 23rd, 2017|Courses, First year, MAM1000, Uncategorized, Undergraduate|7 Comments

Philosophy of Mathematics, by Øystein Linnebo – A review, by Henri Laurie

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.…

By | August 21st, 2017|Book reviews, Reviews|3 Comments

Some sum identities

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:

 

\sum_{i=1}^n 1=n

\sum_{i=1}^n i=\frac{n(n+1)}{2}

\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}{6}

\sum_{i=1}^n i^3=\left(\frac{n(n+1)}{2}\right)^2

 

Let’s go through these one by one. We must first remember what the sigma notation means. If we have:

 

\sum_{i=1}^n f(i)

 

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:

 

\sum_{i=1}^n f(i)=f(1)+f(2)+f(3)+f(4)+....+f(n-2)+f(n-1)+f(n)

 

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:

 

\sum_{i=1}^n 1=1+1+1+1+....+1

 

Where there are n terms.…

By | August 20th, 2017|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

MAM1000W 2017 semester 2, lecture 1 (part ii)

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:

 

d=d_1+d_2+d_3+d_4+...d_{12}

 

Where d_i is the distance walked in the i^{th} five minutes. To calculate a distance, we need to know how fast we are going, and for how long. In fact:

 

distance=velocity \times time

 

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.…

MAM1000W 2017 semester 2, lecture 1 (part i)

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:

  • The antiderivative
  • Sigma notation
  • Areas under curves

Antiderivatives

An antiderivative of a function f on an open interval I is a function F such that:

 

F'(x)=f(x) for every x\in I

 

Note that we say an antiderivative, not the antiderivative. There can be many functions whose derivatives give the same thing. While we know that:

 

\frac{d}{dx}\sin x=\cos x

 

And therefore we can say that \sin x is an antiderivative of \cos x. However, we can also say that:

 

\frac{d}{dx}(\sin x+3)=\cos x

 

So \sin x+3 is also an antiderivative of \cos x. In fact for any constant c it is true that \sin x+c is an antiderivative of \cos x. We write this statement as:

 

\int\cos x dx=\sin x+c

 

This is called the indefinite integral of \cos x with respect to x.…