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MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Introduction to Integration

Here is a draft of my Introduction to Integration

I am still trying to work how best to present this, so any ideas or criticism is welcome.

(There may also be mistakes, let me know if so).

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Introduction to Integration

hi MathsIsFun

A good start.  On the linked 'Integration Rules' page, I assume the integration by parts and substitution are still in preparation.

Because the fundamental theorem uses areas and some teachers gloss over 'reverse of differentiation' bit, I have met students who have two misconceptions about integration:

(i)  They may think that integration is defined as the reverse of differentiation;

(ii) they may think you can only use it to find areas.

I like your approach as it makes it clear that (i) is not the case.

To avoid (ii) I wonder if you would consider re-wording this bit:

Integration is a method to find the area underneath the graph of a function like this:

Replace with eg. "Integration can be used in many ways such as to find volumes and to calculate, for example, centres of gravity, but it is easiest to start with finding the area underneath the graph of a function like this: "

This has reminded me that I said I would suggest some more for the double differentiation page.  Sorry, this got shunted down my to-do list, but I'll try to get on with this soon.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Introduction to Integration

Hi Bob

Defining an integral as the area below a graph is equivalent to defining it as the inverse of differentiation and then showing it calculates areas as well.

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Introduction to Integration

Hi Bob, good suggestion.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Introduction to Integration

hi Stefy

Defining an integral as the area below a graph is equivalent to defining it as the inverse of differentiation and then showing it calculates areas as well.

What!  Why have I spent weeks trying to get 21122012 to do calculus properly.

These statements are not the same.

The fundamental theorem shows that if an area function F(x) can be constructed for the function f(x), then F'(x) = f(x).

So starting with F you can differentiate to get f.

But starting with f you do not necessarily get F.  You might get F + C.

So they are not fully inverses operations to each other.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Introduction to Integration

hi MathsIsFun

Maybe that sentence is a bit long.  So it might be better to split it into several shorter ones.

"Integration is a method for adding up infinesimal 'slices' to calculate a whole. 

It can be used for many things eg. calculating a volume or working out a centre of gravity. 

It is easiest to start with finding the area underneath the graph of a function like this:"

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Introduction to Integration

Hi MIF;

Nice page. Very neat and colorful, easy to read. Glad you did not mention that most integrals can not even be done analytically, that scares people.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Introduction to Integration

hi Stefy

Defining an integral as the area below a graph is equivalent to defining it as the inverse of differentiation and then showing it calculates areas as well.

What!  Why have I spent weeks trying to get 21122012 to do calculus properly.

These statements are not the same.

The fundamental theorem shows that if an area function F(x) can be constructed for the function f(x), then F'(x) = f(x).

So starting with F you can differentiate to get f.

But starting with f you do not necessarily get F.  You might get F + C.

So they are not fully inverses operations to each other.

Bob

They are not inverse in the sense a function is inverse.

The Fundamental theory of Calculus states that the area below the graph of a function is equal to the difference of the antiderivative of the function at the two endpoints of the interval on whic you calculate the area...

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Introduction to Integration

I use the word "reverse".

What do you all think of the tap/tank example?

And is it good that I gave an example from the Rules of Derivatives before Rules of Integration, or should I just go straight to Rules of Integration?

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Introduction to Integration

The tap/tank example is a great analogy for integration! Big stuff made out of smaller stuff- it's all Calculus!

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Introduction to Integration

Hi MIF;

You can leave the page the way it is. Unless someone else has found something... Post it!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.