One last Calc 2 Question: Improper Integrals

Question

I got all the way through this problem and the answer seemed legitimate, but I did not get the correct answer. Thank you in advance for any help. I really appreciate it.

 

Find the volume of the solid.

 

The region bounded by the graph of f(x)=2(x+1)^-3 and the x axis on the interval [0,infinity) is revolved around the y-axis.

Answer 1

This is the case of improper integral.

 

Limits M tends to infinite[ integral of 2x/((x+1)^3)dx.     with the limits 0 to M

 

Integration by parts

 

Limits   M tends to infinite[(-1/2)*(2x+1)/(x+1)^2].     with the limits 0 to M

 

Limits M tends to infinite[(-1/2)*(2x+1)/((x+1)^2)].     with the limits 0 to M

 

Limits M tends to infinite [ (-1/2)*(2M +1)/((M+1)^2).            with the limits 0 to M

 

Limits M tends to infinite [ (-1/2)*1/(M+1)].       with the limits 0 to M

 

=0+2*pi

 

 

=2*pi

 

 

 

 

Answer 2

If we set up slices along the x-axis, they'll be cylindrical shells with radius x, height y = 2(x+1)^-3, and thickness dx. The volume of each slice will be 2π x y dx.

2π ∫₀^∞ x 2(x+1)^-3 dx

Move the 2 out, integrate by parts with u = x and dv = (x+1)^-3 dx, and you should get 2π.