## UCT MAM1000 lecture notes part 3

Today is going to be the last day where we will focus on integration by parts. We’re going to run through a few more examples, but each of them will be slightly different from what you’ve seen already and will give you a few more insights into some of the tricks you might have to play.

Definite integrals using integration by parts

Let’s look at the following integral:

$\int_0^{\frac{\pi}{2}} x^2 \sin x dx$

Where as usual we see that the $x^2$ term becomes simpler if we differentiate it and so we choose:

$u=x^2,\,\,\,\,\, dv=\sin x dx$

Thus

$du=2x dx,\,\,\,\,\, v=-\cos x$

Using the normal integration parts algorithm we will get the following, and note that now we need to evaluate the result at the limits of integration:

$\left.-x^2 \cos x\right|_0^{\frac{\pi}{2}}+ \int_0^{\frac{\pi}{2}} 2x \cos x dx$

We can’t quite solve this last integral directly so we integrate by parts again. However, we note that the first term in this expression actually evaluates to zero when we plug in both limits, so we are left with just the integral term.…

## UCT MAM1000 lecture notes part 2

Before getting onto the maths proper today I wanted to discuss why you should care about any of this stuff. Who cares whether you can integrate a given function or not? There are many reasons as to why you might care, but I think it’s nice to take an example from either end of the reasoning spectrum.

The first reason is that within many different sciences, the way we describe the world is by differential equations, which we will come on to in the coming weeks. Differential equations are like algebraic equations (that you’ve been studying for years) but they include derivatives. An example is something like:

$\frac{df(x)}{dx}+x^2+f(x)=2$

Here you are not being asked to solve for a variable and work out what constant it equals, you are being asked to solve for a function to see how it varies over $x$.

Such equations are the way we model the world, in just about every field from physics, to sociology and from actuarial sciences to epidemiology.…

The first rule of bootcamp is: You DO talk about bootcamp!

I know that the start of the semester is busy, but now is the time to consolidate what you learnt last semester if you feel that you are struggling. It will take a good number of hours each week for a couple of weeks, but with effort you can do it. It’s really useful to go through these with other people, but it’s also important to spend some time going through exercises on your own. I suggest finding a balance between the two.

I’m going to list some links below, some of which have explanations, some of which have videos and many of which have exercises. Here’s your task if you choose to accept it:

1. Put aside Whatsapp, Facebook, news feeds, anything distracting and make sure that you are in a quiet place – if possible take the material offline so you don’t have to have any internet connection when you are going through these.

## UCT MAM1000 lecture notes 1 (part iii) – integration by parts

Here we are going to come up with our second method for solving integrals which we can’t solve by inspection (noting that they are the antiderivative of some function which we know well).

We know the product rule for differentiation:

$\frac{d (f(x)g(x))}{dx}=f'(x)g(x)+f(x)g'(x)$

Integrating it gives us:

$f(x)g(x)+c=\int f'(x)g(x)dx+\int f(x)g'(x)dx$

We can then rearrange this to give:

$\int f(x)g'(x)dx=f(x)g(x)-\int f'(x)g(x)dx$

We have dropped the $c$ because this will come also from doing the second integral.

This is also sometimes written in an alternative form:

Let $f(x)=u$ and $g(x)=v$. Sometimes we will keep the functional dependence explicit, and sometimes we won’t – it is up to you to follow what things are functions of $x$ and what are constants. You will get more and more familiar with it over time.

Now we can write:

$f'(x)=\frac{du}{dx}\,\,\,\,\,\,\,\,g'(x)=\frac{dv}{dx}$

Now put this into the equation above for integration by parts.

$\int u \frac{dv}{dx} dx=uv-\int v \frac{du}{dx} dx$

cancelling the factors of $dx$:

$\int u dv=uv-\int v du$

(Actually the idea of cancelling the factors of $dx$ is rather sloppy, but here it goes through ok).…

## UCT MAM1000 lecture notes 1 (part ii) – integration by substitution – review.

This is just a quick reminder. If you find any of this confusing, there is a very important trick for making it easier – practice, practice, practice! It doesn’t take years to master this, it takes a few hours every week for a few weeks. You will become more and more familiar with the techniques and learn intuitively to know which technique to use in which situation.

If $F(x)=x^2$, what is its derivative? ie. how do we find a function $f(x)$ such that:

$f(x)=\frac{d (F(x))}{dx}$

The answer of course is $f(x)=2x$. We use our normal differentiation rules which you should now be very familiar with. How about if I told you that there was some function $F(x)$ whose derivative was $2x$ – ie. we reversed the question:

What is a function whose gradient at a point $x$ is $2x$?

How do you find what $F(x)$ is? We are trying to solve the equation

$2x=\frac{d F(x)}{dx}$

for $F(x)$.…

## UCT MAM1000 lecture notes 1 (part i) – preamble.

I will attempt to post notes for the coming sixty lectures on a daily basis. You can either ask questions about the topics which you don’t understand here, or email me directly, or of course come and chat in my office when I’m around.

• These notes are for the second semester of MAM1000. They are neither complete nor exact and no responsibility is held for the accuracy within. Mistakes are undoubtedly included. That being said, I hope that they can be a useful resource in addition to the course textbook (Stewart) and additional online materials.
• I am always very grateful when people find mistakes in these notes. These may be in the form of spelling, grammar, calculational errors, typos in formulae, typesetting errors and anything else which doesn’t seem to make sense. If an explanation is not clear, please contact me and I will do my best to explain it in another way.

# Attracting women and girls to careers in physics

This seminar may be attended via video conference in Pretoria, Cape Town and KwaZulu-Natal. Details are indicated below.

The HSRC Seminar Series In collaboration with the Africa Institute of South Africa

Speaker: Dr Malebo Tibane, Department of Physics, Unisa,

Discussant: Dr Palesa Sekhejane, HSRC

Chair: Prof. Narnia Bohler-Muller, HSRC

Venues in Pretoria, Durban and Cape Town (Videoconferencing facilities: see below)

Date: 6 August 2015

Time: 12h30 – 14H00

Representation of girls and women in the Physics related sector is declining or stagnating globally and locally. Therefore, issues that hamper on the development of women to participate equally in male-dominated spheres need to be discussed and captured adequately to create an enabling environment for women and girls in the sciences, and physics in general. In November 2005 Women in Physics in South Africa (WiPiSA) was launched, with funding from the Department of Science and Technology (DST) and under the auspices of the South African Institute of Physics.…