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.
This function is badly behaved at x = 1, which is the lower limit of integration. We write the improper integral as the appropriate limit and work out the integral:
As b approaches 1 the quantity (b – 1)1/2 approaches 0. The integral converges to 2.