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#1 2015-03-10 18:17:26

Justin Johns
Member
Registered: 2015-03-10
Posts: 2

What's really integration?

I tried to look at the example over here:Math is Fun it says that when you integrate 2x you get x^2.It's right if you just look it based on theorems.Does x^2 at 3 means adding 2x '3' times in integration?But if you go this way you wouldn't get x^2 at x=3 which is 9 equal to 2*3=6.

If integration is adding I tried to add upto the three values at x=1,x=2 and x=3 but then the value becomes 12(2+4+6) using the equation 2x.How the value 0f 9 really came while integrating 2x at x=3 as shown in the example of filling a tank with water.Could anyone help me to get what really integration is other than saying it's just adding slices together to get a full piece.

Last edited by Justin Johns (2015-03-10 18:25:29)

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#2 2015-03-10 20:40:20

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: What's really integration?

hi Justin Johns

Welcome to the forum.

what really integration is other than saying it's just adding slices together to get a full piece.

Cannot do that because that is what it is.  The integral sign is a fancy S to stand for sum.

In this case:

Try thinking about the graph y = 2x.

From 0 to 3, it is a diagonal line going from (0,0) to (3,6).  The 'area under the line' is the area of a right angled triangle with a base of 3 and a height of 6, so the area is 0.5 x 3 x 6 = 9.

You will never get the right answer just by adding the y coordinates of the whole numbered x values.

In general, it doesn't have to be areas.  Anything that can be written:

can be evaluated using integration.  If you want, I'll show how to work out (for example) the volume of any pyramid (any base shape) using integration.

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 smile

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#3 2015-03-10 23:09:51

Justin Johns
Member
Registered: 2015-03-10
Posts: 2

Re: What's really integration?

That's a good answer.I'm still confused about why we take the area in integration while the derivative just takes the slope of points.Actually why do we need to consider an area under the curve as said here:Integration when we only need to consider the function or more precisley the line or the curve that produced the function but not the points below that doesn't obey with the function.

Last edited by Justin Johns (2015-03-10 23:10:33)

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#4 2015-03-10 23:29:31

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: What's really integration?

hi Justin Johns

Differential calculus and integral calculus are just two tools that mathematicians use.

What they have in common is they both consider what happens to the value of an expression as things tend to a limit.

In differentiation, when you want the gradient on a graph at a point A, you start by choosing a nearby point B and working out the gradient of AB.  Then you let B get closer and closer to A and see what happens to the gradient.  For 'well-behaved'* graphs the gradient gets closer and closer to a fixed value, so it seems reasonable to assume the gradient of the tangent at A is that value.

In integration, you construct an expression for a 'small amount' of something and then add up lots of those to get an approximation for what you're trying to find.  One way is to under-estimate and hence get a lower bound and then repeat with over-estimates to get an upper bound.  If you find that improving the approximations makes the lower bound higher and the upper bound lower then the answer you want gets sandwiched between ever closer amounts and so, again, you can get a value.  Again, the function needs to be 'well-behaved'.*

One surprising, and very useful, result of the two processes is that with one small proviso differentiation and integration are opposite processes.  So if you start with a function and integrate it, then differentiate the result, you get back to where you started.  This is called the fundamental theorem of calculus. You can work out rules for differentiation and use them to work out many rules for integration.

*Some functions are so odd that they cannot be differentiated.  And if you just make up a random function to integrate there's no guarantee that you'll be able to find a function that would differentiate to it, so integration may be difficult or even impossible.

Most calculus courses start with differentiation and then go on to integration.  My teacher when I was doing this said that integration is the opposite of differentiation (as a definition) but that's not really true.  It's a summation process that can be shown to be the opposite.  That's another thing I can show you if you want, but it's probably best to do some examples first so you are familiar with how it all works.

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 smile

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