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
(p – 1) is negative.
Since we're told p ≠ 1, we know that the antiderivative of
can't be lnx, but is
The integral works out mostly the same way it did in the previous problem:
However, things are a bit different for the final steps. Since p < 1, the exponent p – 1 is negative. This means that as b approaches ∞ the quantity bp – 1 is getting smaller and smaller approaching zero, so the quantity
is getting larger and larger approaching ∞. This integral diverges.