P1^
6
Area(^) as
(^) the
(^) limit
(^) of
(^) a
(^) sum
The estimated value of A is
2 + 5 + 10 + 17 = 34 square units.
This is an underestimate.
To get an overestimate, you take the four rectangles in figure 6.8.
The corresponding estimate for A is
5 + 10 + 17 + 26 = 58 square units.
This means that the true value of A satisfies the inequality
34 A 58.
If you increase the number of rectangles, your bounds for A become closer. The
equivalent calculation using eight rectangles gives
1 ++^13825 ++^2985 ++^538172 +<^858 A<+^13825 ++^2985 ++^5381277 ++^85813
39.5 < A < 51.5.
Similarly with 16 rectangles
42.375 A 48.375
and so on. With enough rectangles, the bounds for A can be brought as close
together as you wish.
ACTIVITY 6.1 Use ICT to get the bounds closer.
x
y
y = x^2 + 1
0
1
5
10
17
26
1 2 3 4 5
Figure 6.8