For which positive numbers a now considering all positive

Question

Suppose that 'a' is a positive constant and that 'R' is the region bounded above by

y = 1/x^a,

below by y = 0, and on the left by the line x = 1.

a) Sketch the curves

y = 1/x^a

for a = .5, 1 and 2. Which of these is closest to the x-axis?

b) For which positive numbers 'a' (now considering all positive numbers) do you get a convergent integral when you attempt to calculate the area of 'R'?

c) Same as b), but for the volume of the solid obtained by rotating 'R' around the x-axis.

d) Same as c), but for the volume of the solid obtained by rotating 'R' around the y-axis.