Understanding Engineering Mathematics

• the definite integral interpreted as thearea under a curveas illustrated
  in Figure 10.13.

A

A =

∫

f(x) dx

y = f(x)

y

(^0) a b x
b
a
Figure 10.13Area under a curve.
For theoretical discussion it is the latter viewpoint which is perhaps most useful – indeed,
the indefinite integral can be viewed as the definite integral
∫x
a
f(u)du
First we will show, using only simple ideas of limits, the connection between the two
viewpoints and then concentrate on the area viewpoint.
For a continuous curve, we can regard the areaAunder the curve as a function ofx,
as shown in Figure 10.14.
y
y = f(x)
x x + h x
C
B
f(x) A
0
f(x+h)
a
D
Figure 10.14The indefinite integral as an area.
The area under the curve betweenaandx+his denotedA(x+h). The difference bet-
ween the area under the curve betweenaandx+hand betweenaandxisA(x+h)−
A(x)and this is clearly the area of the thin strip above the interval (x,x+h). The area
of this thin strip is approximately^12 [f(x+h)+f(x)]×h, the area of the thin trapezium
ABCD,ifhis very small. So we have
A(x+h)−A(x)^12 [f(x+h)+f(x)]×h