- an area split into strips
- a curve split into line segments
- a solid body split into very small particle like elements
The rest of this chapter will be concerned mainly with geometrical applications, although
the Applications section contains some examples from mechanics. The main objective is
to illustrate the general ideas and motivate the hard work necessary to master integration.
The area ‘under’ a curve is the, perhaps misleading, term used for the area enclosed
between a given curve and thex-axis. Or, it may be the area enclosed between two quite
general curves. Figure 10.12 illustrates the sorts of possibilities we can have.
(i)(iii)(ii)(iv)Figure 10.12Areas and curves.
The case (i) is the easiest to deal with. In case (ii), note that areasbelowthe axis are
regarded asnegative. This is to conform with such results as:
∫ 0
−π/ 2sinxdx=− 1In case (iii) we would need to find where the curves intersect and integrate the differ-
ence between the two functions over the appropriate region. Case (iv) also requires us
to integrate the difference of the functions, but now care must be taken to allow for the
difference in signs of the two areas.
Considering now the simple case (i) let us look at the connection between integration
and the area under a curve.
At the elementary level there are two common viewpoints of integration –
- the integral as the inverse operation to differentiation, oranti-deriva-
tive(253
➤
):
∫
2 xdx=x^2 +Cbecause
d
dx(x^2 +C)= 2 x