
11-6 Calculus – Differentiation and Integration
Next, in a similar manner, we take the greatest value of f(x) in each subinterval and construct a
rectangle with that as its height as shown in Fig. 11.4. Here, we observe that the sum of the areas of
these rectangles is greater than the area we are required to find. This sum is called upper sum.
Thus, it is understood that the area we need to find lies between the upper and lower sums which can
easily be calculated as these are sums of areas of rectangles.
Let the greatest value of f(x) in the ith subinterval be
f x
i
( )
*
and the least value be f x '
i
( ) (Fig. 11.5).
Thus, the upper sum is given ...