When is the volume finite? Let R be the region bounded by the graph of \(f(x)=x^{-p}\) and the x-axis for \(x \geq 1\).
a. Let S be the solid generated when R is revolved about the x-axis. For what values of p is the volume of S finite?
b. Let S be the solid generated when R is revolved about the y-axis. For what values of p is the volume of S finite?
Problem 62EWhen is the volume finite Let R be the region bounded by the graph of f(x)= and the x-axis for x 1.a. Let S be the solid generated when R is revolved about the x-axis. For what values of p is the volume of S finiteb. Let S be the solid generated when R is revolved about the y-axis. For what values of p is the volume of S finiteSolution Step 1:(a)The volume of S is given by the integral If -2p = -1 (i.e. p = ½) , then this integral becomes = log aAnd the volume is infinite, If p , we have: = = = -
