7.2 Definite Integral as an Area under a Curve
Observe the adjoining figure. It is clear that f (x_{r}) Dx_{r} is the area of the rectangle with base Dx_{r} = x_{r}  x_{r}1 and height f (x ^{r} ).
If now n ® ¥ , in such a way that the largest of all the sub intervals, Dx_{1}, Dx_{2}, .......Dx_{n} tends to zero, then we should expect that Sn would approximate to the area say A, bounded by the curve y = f (x), the x  axis and the ordinates x = a and x = b.
That is lim Sn = A, provided the limit exists .
n ® ¥
Example 1 Evaluate the Riemann Sum for f(x) = x_{2} on [1, 4] using 6 sub intervals of equal length.
Solution : Since the sub intervals are to be of equal lengths, you find that
