Solution Found!
When is the volume finite Let R be the region bounded by
Chapter 7, Problem 61E(choose chapter or problem)
When is the volume finite? Let R be the region bounded by the graph of \(f(x)=x^{-p}\) and the x-axis for 0<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 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 ?
Questions & Answers
QUESTION:
When is the volume finite? Let R be the region bounded by the graph of \(f(x)=x^{-p}\) and the x-axis for 0<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 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 ?
ANSWER:Problem 61EWhen is the volume finite Let R be the region bounded by the graph of and the x-axis for 0 < 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 finiteSolutionStep 1In this problem we have to find when then the volume is finite.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 finiteThe volume of S is given by the integral If (that is ) then the integral becomes, If , we have Thus the volume is finite if .Hence if p > 1/2 the volume is finite, but if p 1/2 the volume is infinite.