Integration Skills &#=
8211;
Elemental Area & Volume
Example 1:
Imagine you have a circle of radius r as shown below. Now, lets make the circle slightly bigger so =
that
the radius is (r + dr). Notice that the area also inc=
reases
shown shaded.
This shaded area is known as an “elemental
area”. Elemental area and volume are very useful when doing integrati=
on.
If we assume that dr
is very small, and straighten the elemental area, it will look almost
rectangular as shown below:
Hence elemental area, dA
=3D 2=
pr dr
Using this elemental area, we can derive the total are=
a of a
circle. Imagine we start with r=
=3D 0,
then keep on adding more elemental area until the radius is r. What we’re doing is actua=
lly to
integrate the elemental area from 0 to r,
ie.
![](3D"ElementalVolume_files/image005.gif")
That wasn’t so bad, was it? Now let’s exte=
nd
this concept to elemental volume.
Example 2:
Again, imagine you have a sphere of radius r. We know that the surface area o=
f the
sphere is given by 4pr2.
Increasing the radius by dr will
generate the following elemental volume : (Come on, stretch your imaginatio=
n a
little!)
Hence elemental volume, ![](3D"ElementalVolume_files/image008.gif")
And to derive the total volume, we just need to keep on
adding (ie. integrate) the elemental volume from 0 to r.
![](3D"ElementalVolume_files/image010.gif")
as expected=
!
Now, go forth & integrate!!!
Prepared by: &nbs=
p; Mr
Faizal, 11 Jan 2005
For: =
&nb=
sp; 2005
NJC JC2 S-Paper Resour=
ce
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