History & Philosophy of Calculus, Session 4 THE INTEGRAL.

History & Philosophy of Calculus, Session 4

THE INTEGRAL

In this session we meet integration, which in a sense is one half of the thing called calculus. In the next session we meet the other half, so at that point we will (at last) have explained what calculus is.

The groundwork we covered in the previous three sessions will start to be useful to us:Archimedes’ infinite approximation of the area inside a parabola.

Analytic geometry and the connection between curves and algebra.

CALCULUS, AT LAST

SPEED-TIME GRAPHS REVISITED

TACHOGRAPHS

• Tachographs are used to make sure commercial drivers stick to speed limits and take proper rests.

• It’s easier to record speed in a car than distance, so for our purposes we’ll imagine we have a record of a journey in the form of a speed-time graph.

GRAPH #1

y = 30

GRAPH #2

y = 30

GRAPH #3

y = x/30

GRAPH #4

The motion described in all these examples can be captured by some kind of algebraic equation with two unknowns: speed and time.

Remember what Fermat realised: every equation in two unknowns corresponds to a path in the plane – a curve, which we usually call the graph of the equation.

If you have a speed-time graph, we’ve seen that you can calculate distance by finding the area between the curve and the x- and y-axes. This insight is only possible with analytical geometry!

Speed and distance are basic to dynamics – the branch of physics that studies motion. This includes everything from astronomy to engineering and was the main motivation for the development of the calculus.

THE SPEEDDISTANCE RELATIONSHIP

Really we didn’t use any information about distance or speed in any of this, just a relationship:

Speed is the rate of change of distance over time.

That means the same methods can be applied to anything that has a rate of change.

Sometimes (as with the tachograph) we know the rate of change and want to find how much has changed.

For this, a general method of finding areas under curves is needed.

This general method will be called integration, and is one half of the calculus.

RATES OF CHANGE

Speed

Distance

Area Slope

Rate of change of

thing

Thing

Area Slope

GRAPH #5

?y = -25x² + 50x + 10

CAVALIERI’S RULE

A WARM-UP: JOHN WALLIS’S INFINITE FRACTIONS

“Since, moreover, as the number of terms increases, that excess over one thirdis continually decreased, in such a way that at length it becomes less than anyassignable quantity (as is clear); if one continues to infinity, it will vanish completely.”-- John Wallis, The Arithmetic of Infinitesimals, Prop 20.

We’ve seen how Archimedes found the quadrature of the parabola using synthetic techniques. But there are two problems: It’s not clear whether his arguments are robust. His argument is specific to the parabola. Not every curve in

real life is either a straight line or a parabola, and we don’t want to have to find a new formula from first principles every time.

What we need is a more general method.We’ll stick with the parabola for a moment and look

at a method similar to the ones used by Cavalieri and Fermat around the 1630s. These use analytical techniques and a bit of algebra, and it

turns out that makes them easier to generalise. You still might wonder about whether these arguments

really “work”, though – this is something we’ll come back to.

CAVALIERI, FERMAT AND WALLIS

We’ll look at the area under the parabola y = x2 between x=0 and x=5.

The argument will generalise right away to any parabola between any values of x, but it will also cover many other curves if we’re careful.

OUR EXAMPLE

DIVIDING INTO RECTANGLES: STEP 1

DIVIDING INTO RECTANGLES: STEP 2

WHAT IS THE ESTIMATE?

• 1x12 + 1x22 + 1x32 + 1x42 + 1x52

• Note that the width of each rectangle is 1 unit.

It’s geometrically obvious that if n increases from 5, the rectangles are smaller relative to the area being calculated and so the estimate improves.

Cavalieri tried the following trick: he tried to find out what happened to the ratio of the rectangles to an easy-to-calculate area as n increased. This, he thought, would give him a better and better estimate of the real area.

CAN THE ESTIMATE BE IMPROVED?

THE COMPARISON RECTANGLE

The area of the green rectangle is always n x n2, which is n3, regardless of the value of n.

A BIT OF ALGEBRA

• This is the formula for the sum of the squares of the first n whole numbers:

• This wasn’t new; it had been known to Indian mathematicians since at least 500CE.• Much Indian algebraic knowledge passed

through the Arabic world into Western Europe.

We now have the ratio:

In (modern) symbolic terms:

After some simplification:

Now the crucial step – as n gets very big, the second and third parts get “as small as you like” leaving, for large n,

SOME MORE ALGEBRA

So we have:

Rearranging:

The big win here is that we know how to calculate the green rectangle’s area, so we now have a “closed form”:

This is a purely algebraic expression of what Cavalieri believed to be a geometric fact about the area under the parabola.

SOME MORE ALGEBRA

CAVALIERI’S CONCLUSION

The area under the curve y=x2 from x=0 to x=n is equal to

It’s pretty easy to generalise this idea to any curve whose equation is y = x k for some whole number k. If you enjoyed the algebra on the previous few slides, try to

prove it for yourself!This leads to what’s sometimes still called Cavalieri’s

Rule. In modern notation:

The left-hand side of the “=“ sign can be read as “The area under the curve y=x k from x=0 to x=n”.

Later this will be called the integral of xk from 0 to n. A special case of Cavalieri’s Rule says that the integral of a

constant k is kx. We’ll see an example with this in a moment.

CAVALIERI’S RULE

FROM CAVALIERI’S RULE TO A CALCULUS

Cavalieri’s Rule is very nice, but is it just another special case?

It’s inefficient to have to keep doing this much work every time we meet a new curve – now we’ve done the hard part we’d like to be able to convert it into a “recipe” that can deal with many different situations easily.

This is exactly what a calculus is!These methods came quite quickly, and were

invented by multiple people at once in the middle of the seventeenth century.

They enable us to deal with a large class of polynomials – expressions involving powers of x. These accounted for most (but not all) of the curves that were known to mathematicians at that time.

THE NEED FOR “A CALCULUS”

Suppose we just want to find the area between x=2 and x=5.

This isn’t too difficult:

So if this was a speed-time graph of a journey with speed in mph and time in hours, the distance travelled would be 39 miles.

MORE APPLICATIONS

These were worked out slowly, by many people at the same time. Also remember we’re presenting them in modern notation that arrived quite a lot later.

In these, suppose f(x) and g(x) are just algebraic expressions involving x and some numbers.

The Constant Multiple Rule:

The Sum Rule:

These, plus Cavalieri’s Rule, allow us to integrate a curve of the form y = f(x) where f is any polynomial at all! This is an amazing improvement over Archimedes’s result.

SOME ALGEBRAIC RULES

Let’s try a more complicated one.

Here’s the curve of the equation

We’d l ike to find the area under it between x=1 and x=3. This would have been very

difficult, maybe intractable using synthetic techniques.

Although the calculation on the next slide has a lot of steps, each one is easy and mechanical.

EXAMPLE

Sum Rule:Constant Multiple Rule:Subtracting areas:Cavalieri’s Rule:The rest is just arithmetic, giving the answer:

EXAMPLE

What you’ve just seen is an example of the power of the calculus.

We dealt with a problem we’d never seen before, and although it involved quite a lot of symbol-juggling, it wasn’t really all that hard and didn’t require any clever new ideas.

We teach 16-year-olds to do what we on the previous slide in the first year of A level maths. We can do this because it’s a calculus: it’s been turned into a systematic method for solving a large number of problems.

But this raises a question: is it reliable? Methods involving “going to infinity” are questionable, and as Zeno

reminds us they can lead to wrong conclusions. Cavalieri may have got reasonable-looking answers in the cases he

tried, but what if sometimes it gives the wrong answer? Can we be certain of the results of calculus, the way we (think we)

can of Euclidean geometry? Or are we in the realm of mere approximations?

THE INTEGRAL CALCULUS