How do you introduce Integral Calculus, I've never found a way I'm happy with

Discussion in 'Mathematics' started by ian60, Nov 9, 2010.

1. ian60New commenter

I will be introducing Int Calc soon (this is an IB HL course)
For some reason I don't want to just do 'anti-differentation'. Am I wrong?
Is there an easy to follow, 'first principles' approach, that others have used?

Eventually (ie in a week or two) I'd really like to explain <u>why</u> the integral between limits gives the area. Easily done?

I know this may show me to be an idiot within the Maths Community, but you have always been very helpful in the past.

2. ian60New commenter

I will be introducing Int Calc soon (this is an IB HL course)
For some reason I don't want to just do 'anti-differentation'. Am I wrong?
Is there an easy to follow, 'first principles' approach, that others have used?

Eventually (ie in a week or two) I'd really like to explain <u>why</u> the integral between limits gives the area. Easily done?

I know this may show me to be an idiot within the Maths Community, but you have always been very helpful in the past.

3. hassan2008

I do not use any first principles approach.
I use Autograph to introduce integration and its successful.

4. ian60New commenter

Thank you hassan2008, that sounds similar to whatI would normally do (I use GSP rather than AutoGraph, for no real reason).
I was just wandering if any folk out there had a different approach.

For a few examples that are similar to how you demonstrate differentiation from first principles, you can uses the summation formulae for various powers of n. i.e. stuff like n(n+1)/2. Divide the interval from 0 to x into h equal parts and let h tend to 0.

I don't know if this is too sketchy for you, but I think you should find this approach in quite a few textbooks.

6. Polecat

I would show that the derivative of the area is the function, so the anti-derivative of the function with limits gives the area. Then say that most people call the anti-derivative the integral because it is less of a mouthful.

7. PaulDGOccasional commenter

It may help to introduce "real life" integrating devices.
The odometer is an obvious example.

8. ian60New commenter

Good stuff from all of you, thank you for helping.

9. AnotherMathsHoD

I've done the integral of x^2 between limits from first principles by considering the area in terms of the trapezium rule and then taking the limit as the width of the trapezia tends to zero (and the number of trapezia tends to infinity)... It's a bit clumsy in places but it works out.
It's one of those things I like to do with a good group in a lesson where half the class are away for some reason...
I don't introduce it that way though I do it as anti-differentiation first then, do the area argument by starting with the partial area and adding on a trapezium whose area lies between y x delta x and (y + delta y) x delta x, which means that deltaA/deltax (where deltaA is the change in Area) is between y and y+ delta y and taking the limit of this as delta x tends to zero gives that dA/dx=y which using the antidifferentiation argument gives A as the integral of y...
That's just a rough overview but I hope it gives you an idea - there is a good explanation of this in the Cambridge IB Higher Level textbook by Douglas Quadling...

10. oscars

Is this powerpoint any use?

12. Betamale

I think it depends of your students.
The edexcel SOW/books often approach it with far less depth and end up feeding you an algorithm...shame as some good pupils might want to go further.
Im a being a pedant by asking if we are correct in using the word 'antiderirvative'?

13. Maths_MikeNew commenter

antiderivative is and acceptable alternative to indefinite integral - but even if it was just a made up something that the students understood to mean the opposite of derivative would it matter?

14. weebecka

yes it matters - if there was a different official word it would be useful to know that too!

Anyway I find antidifferentiation is the best mental image for indefinite integrals, but the summation of area is the best menatal image for definite integrals.

Oscars - I was really surprised the powerpoint you recommended didn't bother to do any real calculations of areas under functions with rectangles. It just went straight to generalised algebra. I find the imagery of splitting the with of the definite integral into equal areas, drawing rectangles the height the function is in the middle of each interval and calculating and adding the areas of the rectangles to be a useful intermediate step. Then the generalised algebra fits on top of rich visual imagery and practical experience. But it's tedious calculation so I usually bang together a spreadsheet with them to facilitate it (they have to do sketches). I've just done one to illustrate what I mean and you can get it through this link:
This particular one is set up at the minute to calculate the area under y=x^2 between 2 and 3 with 30 rectangles, but it's easy to adjust - just change the figures in the blue squares and the final summation formula.

I always try and at least show students an approach to differentiation and integration from first principles.

Especially at A Level I make it very clear that it is not all about "finding areas" or "finding gradients" but that they are general techniques which can be applied to a massive range of problems in maths, physics etc.

I don't want them coming away from Core maths thinking its all about tangents and areas, or coming away from C1 thinking all they have to know is "bring the power down and lower the power by 1".

Regardless of how much of it will be tested in the exam, if they fail to understand the principles behind it, or the reason why we use certain notation, or how to use that notation, or the difference between a "standard result" and a "property" of the technique, they will struggle greatly as they progress through maths. In my experience the great drop off that occurs around C4 is just down to poor foundations being laid at C1/C2.

It IS worth time invested in some approaches from first principles and for those students who enjoy extension work that will include an example in differentiation which is NOT about a tangent or normal to a curve, or an example that is NOT about the area underneath a curve. If only to drum home how versatile these skills can be.

It is also worth a bit of time looking at limits as this is not formally taught at all at A Level (and yet is relied upon in differentiation, integration and series!).

I normally make up a booklet of my own and lead them through it, so they can concentrate on the key points without having to copy anything, have something to mull over at home a few times and also allows them to refer to it when I pull them up on how they are writing their working (explaining why and how we use certain symbols).

An hour spent here at this level can save 10 hours of hair tearing at C4.

16. weebecka

It looks like you have the same formatting issue as me.

Before a paragraph you have to type 3 characters: less than, p, more than (if I used the symbols I would get a new paragraph instead of showing you what to do). After a paragraph it's 4 characters: less than, p, /, more than.

17. ian60New commenter

I asked my lot what is meant by 'anti-pasta'. Most of them hadn't a clue, but we got to the idea of 'something before the pasta dish'.
After this it wasn't a huge leap to ask them what an anti-derivative might mean. Though they did complain about being hungry.

18. Polecat

Antiderivative is correct, as is the term 'primitive'.

19. florapost

i've had a long fortnight and i'm tired - this is a joke and you do know the difference between anti and ante, right?
is this the secondary equivalent of all that hunger-inducing pizza and cake dividing when doing fractions in primary?
btw - how was parents evening?

20. florapost

i take that back - i am wrong and i am rubbish
and i am tired!!!!