Volume of a Cylindrical shell: xy=1, x=0, y=1, y=3, Calc I help

Find the volume of the Cylindrical Shell (don't use washer method)

xy=1, x=0, y=1, y=3,

Additional Details
Answer is 4pi, show the way plz.

Rotating About X Axis

Since we are using cylindrical shell method and rotating about x-axis,
we will integrate with respect to y:

xy = 1 ----> x = 1/y

Radius of cylindrical shell = y
Height/length of cylindrical shell = 1/y

V = 2π ∫ₐᵇ r * h dy
V = 2π ∫₁³ y (1/y) dy
V = 2π ∫₁³ 1 dy
V = 2π y |₁³
V = 2π (3 − 1)
V = 4π

height=1/y
radius: y+1
2pi*INT[1/y(y+1)dy]
2pi*INT[1+1/y dy]
2pi*[y+lny]