C1

Integration

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A

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Among the Greek mathematicians, two stand out in my book.

One of them is Euclid, the genius behind the greatest mathematics book ever The Elements which is the most successful and influential textbook in history and I think I'm right in saying that it is the third most widely read book ever. Only religious books have been read by more people.

But he's not the Greek fella I'm getting to. I am thinking of Archimedes.

Archimedes...

• Had his Eureka moment when he discovered that when something sinks, it displaces its own volume. Imagine a pool of water. Floating on it is a boat with a heavy ball inside. If the ball is now taken from the boat and dropped into the pool, does the water level rise, fall or stay the same? The answer is below
• Invented the Archimedes screw (look it up) which is still being used today.
• Advanced human knowledge of levers. His catchphrase was "Give me a place to stand on and I will move the Earth."
• Predicted the number of grains of sand that would fill the entire Universe, in doing so he invented a sort of standard index form, way before the Hindu-Arabic number system arrived in Europe
• Came up with a way of calculating sqaure roots
• Figured out a pretty decent approximation to π
• But his best work was to calculate the area of a circle. No, really, imagine what a breakthrough this must've been.

When the ball is in the water, it displaces the same volume as itself. When the ball is in the boat floating on the water, it displaces it's own weight of water. The ball sinks so it must be more dense than water. There is more water displaced by the ball's weight than by its volume so the answer is that the water level goes DOWN when the ball is in the pool.

So what's so good about that. And what's it got to do with integration?

Well, the method he used was to cut the circle into millions of equal slices and rearrange tham to make (nearly) a rectangle.

In doing this, Archimedes was the first person to do what we now call integral calculus.

Where differentiation is all about rates of change (ie gradients) by looking at two very close points, integration is about finding areas (and volumes) by imagining the awkward shape to be made up of an infinite number of infinitisimal strips (much as Archimedes did). In C1 however we will think of integration as the opposite of differentiation.

It certainly isn't obvious that the oppostite of finding the gradient should be to find an area. we will look at this in more detail later (although it is beyond A-level and is not part of this course).

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Indefinite Integration

Indefinite integration is just another word for anti-differentiation.

In other words, in C1 integration is just the opposite of differentiation.

Let's have a look at the notation...

∫ 2x dx, is read "the integral of 2x with respect to x." Which means: "what do you differentiate (with respect to x to get 2x?

And we know from differentiation that the answer is x².

...or x² + 1... or x² + 43... or x² - 6298...

There are infinitely many functions that when differentiated give 2x. They are all x² ± something, so we write the answer as x² + C, where "C" is called the constant of integration and could be any number, positive or negative or zero.

I could give you a clue as to what "C" could be. I am thinking of a function f (x) that, when differentiated, gives f '(x) = 2x. Furthermore, when x = 2, f (x) = 9. What is my function?

We know that the family of functions that differentiate to give f'(x)=2x are f(x)=x^2+C Substitute the values in the "clue" gives: 9=2^2+C and so C=5. The answer is f(x)=x^2+5.

By the way, this is called indefinite integration because we don't get a fixed number as the answer. Remember what I said about integration being about finding areas? Well, areas are fixed numbers and shouldn't involve variables like x. You will learn about definite integration in C2.

Remember that all we can differentiate/integrate must look like axⁿ, where a and n are numbers.

Let's have a look at a couple of exam questions - one easy, the other more involved.