## Beginning Calculus – Integration

What is integration ?

Well in two variables, integrating is something you do to a function over a given range to generate a new function which will give us the area under the curve of the function..

lets think about areas under lines: its easy to find the area under a simple graph like y=5 between 1 and 8

its just a rectangle and is thus 7 x 5 = 35 units

but now lets look at a curve…start with y=x^2

how do we get this area? because its curved our calculation looks like it might be harder.

well here’s how we proceed…

what if we divided the area into small rectangles ?

we could then add up the areas of each rectangle and get an estimate. Fine. but this is a powerful idea we’ve discovered. Calculus deals with the idea of limits. what if our limit was the limit of the sum of the areas of the small rectangles, as the width of each rectangle gets smaller and smaller i.e. it tends to an infinitesimal amount ?

then we can do our calculus trick – the rectangles become infinitesimally small and as they get smaller and smaller the area they sum to fits the curve closer. Thus the limit of this process is our area.

so what is our approximation to the area now? The area of any given rectangle is the width times the height. The height is the y value and the width is our small quantity we’ll call delta x.

the sum of the areas of all the rectangles is

which can be expressed more succinctly using the sigma notation as:

well, as we let delta x tend to zero

we change the summation sigma sign to an integral sign and write:

the fundamental theorem of calculus states that integration is the opposite of differentiation

so if

integrating both sides gives

so

hence over the range specified we write

which works out to

in fact

and since

we can now integrate any polynomial !

Just passing by.Btw, your website have great content!

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MikeMarch 1, 2009 at 11:23 am