Pythonism

code and the oracular

Beginning Calculus – Integration

with one comment

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

straightline

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

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

integral1

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 ?

integral2

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

z

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

riemann

well, as we let delta x tend to zero

lim

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

def-integrw

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

so if

b

integrating both sides gives

c

so

d

hence over the range specified we write

def-integr1

which works out to

sum

in fact

power1

and since

power21

we can now integrate any polynomial !

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Written by Luke Dunn

February 28, 2009 at 4:06 pm

One Response

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

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    Mike

    March 1, 2009 at 11:23 am


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