Example 6 Suppose we want to evaluate Here, f: x Therefore,
+
Defidte Integral
i2'
(X + x2) dx
x + x2is an increasing function on [1,2].
( x + x 2 ) d x = lim h T f ( l + i h ) . h = l / n h+O
lim h h+
O
x
[(l
i=l
+ i h ) + (1 + i h ) '
]
i=l
Recall that
lim [2h h+O
5 1+ 3h2 5 i + h3 x i2] =I
i=l
a=l
3 1 lim [2nh+- h2n(n+l) + - h3n(n+l) (2n+l)] 2 6
I~+O
3 lim [2 + - (1 + h ) o 2
h+
1
+ (1 + h ) 6
(2
+ h ) , since&=
1.
3 1 23 =2+-+-=-. 2 3 6 In this section we have noted that a continuous function is integrable. We have also proved that a monotone function is integrable. corollary 1 gives us a method of fmding the integral of a monotone function. One condition which is very essential for the integrability of a function in an interval, is its boundness in that interval. If a function is unbounded, it cannot be integrable. In fact, if a function is not bounded, we cannot talk of Mi or mi, and thus cannot form the upper or lower product sums. Now on the basis of the criteria discussed in this section you should be able to solve this exercise.
E E10) State whether or not each of the following functions is integrable in the given interval. Give reasons for each answer.
x + l when x < O f(x) =
in [-I,]] 1-x when XSO,