624CHAPTER 10SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALSThis comes about from settingarea of=limb→∞b1dxxp=∞1dxxp.Forp=1,∞1dxxp=limb→∞b1dxxp=limb→∞11−p(b1−p−1)=1p−1,ifp>1∞,ifp<1.Forp=1,∞1dxxp=∞1dxx= ∞,as you have seen already.For future reference we record the following: Letp>0,(10.7.1)∞1dxxpconverges forp>1 and diverges forp≤1.Example 3We know that the region below the graph off(x)=1/x,x≥1, hasinfinite area. Suppose that this region with infinite area is revolved about thex-axis(see Figure 10.7.3). What is the volumeVof the resulting solid? It may surprise yousomewhat, but the volume is not infinite. In fact, it isπ. Using the disc method tocalculate the volume (Section 6.2), we haveV=∞1π[f(x)]2dx=π∞1dxx2=πlimb→∞b1dxx2=πlimb→∞−1xb1=π·1=π.