Write the volume of the solid?

Question

The infinite region in the first quadrant bounded by the coordi-nate axes and the curve y=1/x-1 is revolved about the y-axis to generate a solid. 

 

Write the volume of the solid as an improper integral. 

 

Answer: pi*∫ dy/(y+1)^2 from 0 to infinity

 

Please show all your work.

Answer 1

Use disk method.

 

V=∫pi*r²(y)*dy, where r is the radius of the disk. In this case r=x or r=1/(y+1), hence

 

V=pi*∫₀^∞ dy/(y+1)²

Calculation will yield V=pi