C2: Integration
In C1 we did some anti-differentiation and gave it a name (integration) and a symbol (∫). Yet there seemed to be ne real point in it. Just what is integration and why is it one of the most important areas of mathematics?
Definite Integration
I have placed all theory in its own section "The Fundamental Theorem Of Calculus", so I'll get to the applications here.
Where indefinite integration was simply anti-differentiation, definite integration has a particular point to it. This point is really what integration is all about: finding areas under curves.
So, if differentiation concerns itself with rates of change or gradients, integration is about figuring out awkward areas. There is no obvious connection between the two and it took the genius of Isaac Newton to demonstrate that they were part of the same "calculus" package.
Area Under A Curve
So, what do we mean by "area under a curve"? Here's what.
By the Fundamental Theorem of Calculus we know that integration is the inverse of differentiation. So to find the area under a curve all we need to do is find the anti-derivative and chuck some numbers in.
Here's a few exam questions. They're taken from Jan 2009, Jan 2005 and May 2006. I will use the same exams in the next sub-section on Trapezium Rule.
The Trapezium Rule
You might have realised that, although integration can get quite hard, we can't really integrate many different types of function. For example, try integrating 1/x.
We need a numerical method for integrating more complex functions. Think of this as a bit like "trial and improvement" only at C2 level. We will find the area by splitting it up into several (usually about 5 or 6) strips and find the areas by thinking of them as trapeziums. This is called the Trapezium Rule.
I won't bother going into detail, so I'll dive straight into a couple of exam questions.