Determine whether the integral converges or diverges. Indicate on a graph the region whose weighted area is given by the integral. If the integral converges, find its value.
for p < 1
This integral is also badly behaved at x = 0. Most of the work is the same as in the previous problem. We don't need to change anything until we get to this step:
At this point, since p < 1, the exponent p – 1 is negative. As b approaches 0 the quantity bp – 1 will approach ∞, and so the term
will approach 0. The limit converges to
Since p < 1 the quantity is positive, which is reassuring since the area between and the x-axis on (0,1] is all above the x-axis.