Calculus demystified Part 3

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We wonder, what in the world! was Issac Newton thinking?


NEWTON

"Sir Lord...If you know me by know, it was the right approach from my own thinking"


LORD

We just found out that you have some type of Aspergers, My 'Sir Newton'

NEWTON

I don't understand you at all!, even your accent is different Sir Lord. Asperberger? Who is that?

LORD

Sorry! Never mind..! We want to get your approval in demystifying this matter, once and for all, My Lord Isaac...!

NEWTON

From our optical point of view and if the celestial bodies approve...alas! We will see, Sir Lord!

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BASICS OF INTEGRATION

Imagine you have to fence up your backyard with pieces of wood. The catch is that you have to make them fit on the curve shown on the graph(in blue). You know that every piece of wood(vertical slabs) has an individual area of

A1=Y1 H1

Now, common sense tells you that the total area of that fence will be obtained adding the individual areas which will be represented by the summatory on the graph. But, the problem is, you really didn't cover the whole area smoothly for the curve F(X)=Y. That's where our Math wizards from Mesopotamia, and Egypt, and seventeenth century Europe forefathers tried to solve. Enter Integration '∫'...Now off the record, we suspect that Leibniz created the symbol for integration by accident, since Latin was in vogue. Newton did win on this election of the ∫ symbol for Integration; Could've been I as in integrals, but we can tell you Sir Newton, that if you mend the two first letters of your name, you will see what we are talking about. "I'm certainly impressed Sir Lord. Sleek and clever!"



INTEGRATION

We want you to know... that from now on, the inverse of integration is called differentiation. So we are already linking both math tools by our statement. From the graph above, lets make up to what our teachers told us: We chose an interval 'a' and 'b' for x, then if we represent f(x) as the curve in blue. We can 'integrate' the and finally reach out to the formula: But lets state it in a mathematical form:

We are given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral. (indefinite integrals for a later hub)summatory

∫f(x)dx

is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.

We introduced finally the main purpose of this scary representation. This reminds us of the + sign when we were seven years old, right?

A function F whose 'derivative' is this famous f(x) is called indefinite integral and is represented by

F=∫f(x)dx

Now, take it easy on that. We want you to think...that, "dx is the one Lord told us about in Differentials," so somehow Integration and differentiation are connected, we just need to look for that dy in that tricky F.

if f is a continuous real-valued function defined on [a, b] called close interval, then, once an anti-derivative F of f is known, the definite integral of f over that interval is given by the graph up above.

RIGHT FRESH TO AN EXAMPLE:

The antiderivative of x1/2

Can be found in well known tables but for now is like this: F(x) = 23x3/2

We know that ∫xn dx=(xn+1) /n+1 +C where C is a constant

We simple do the operation for intervals[0,1]

F(1)- F(0)

2/3-0=2/3

This video will clear things better...

We hope we have given you some head start on this important subject for a future Engineer or a Scientist....we will continue on a Part4, where we will work on integration by parts.

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Comments 8 comments

Gypsy Rose Lee profile image

Gypsy Rose Lee 4 years ago from Riga, Latvia

Now I know I'll have to look at that YouTube several times but it doesn't matter. You did a terrific job.


Lord De Cross profile image

Lord De Cross 4 years ago Author

Oh Sir Newton, Mylord! Believe it..or, Certainly Gypsy Rose must've loved your kind of Math! She read our 3 hubs like nothing! Appreciative hearts shall commend this lady upon reaching that star of knowleldge.

LORD


Gypsy Rose Lee profile image

Gypsy Rose Lee 4 years ago from Riga, Latvia

Now you have to take a look at my comment on #2.


JT Walters profile image

JT Walters 4 years ago from Florida

Oh D,

I am melacholy for my university days. Although I had I remember trig integrals to be most challenging. This is an excellent explanation of algebraic integrals.

I miss the glory years of spending my days working these equations. You couldn't think of anything else or worry about anything else but the problem at hand.

Excellent job D.

JT


Lord De Cross profile image

Lord De Cross 4 years ago Author

Thanks again GypsyRose!

Hi JT Walters!

We know what you meant..those days when Thermo and fluid Mechanics were a common thing...don't forget strenght of Materials and the whole nine yards! TRying to work on your trigo Integrals, but we only have 24 hrs a day...and time goes like a train in Japan! Glad to hear from you again JT.

lord


iamaudraleigh 4 years ago

Ok Lord...I do not know anything you were saying. However, you were clever and witty wile formatting this piece perfectly! Voted up and shared!!!


Lord De Cross profile image

Lord De Cross 4 years ago Author

That was nice! Audra! Writing this piece for you... and you just keep fixing that diaphragm from your Cannon Camera!! (JOKING)

Hope to see you smiling as you did before.

LORD


iamaudraleigh 4 years ago

LOL!!!

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