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 positive.
Since we're told p ≠ 1, we know that the antiderivative of
(if p = 1 were an option, the antiderivative could be ln x instead). So when we work out the integral, we get
Since p > 1, the exponent p – 1 is greater than 0. As b approaches ∞ the quantity bp – 1 will grow larger and larger, so
will approach zero. This means the integral will converge to
Since p > 1 this quantity is positive, which is reassuring since the integral refers to an area above the x-axis.