ANSWER
By simplifying the left hand side using the Pythagorean Identity.
EXPLANATION
The given identity is
Take the left hand side and simplify to get the right hand side.
Collect LCM for the denominators.
Recall the Pythagorean Identity.
This implies that:
We substitute this to get,
The disk method will only involve a single integral. I've attached a sketch of the bounded region (in red) and one such disk made by revolving it around the y-axis.
Such a disk has radius x = 1/y and height/thickness ∆y, so that the volume of one such disk is
π (radius) (height) = π (1/y)² ∆y = π/y² ∆y
and the volume of a stack of n such disks is
where is a point sampled from the interval [1, 5].
As we refine the solid by adding increasingly more, increasingly thinner disks, so that ∆y converges to 0, the sum converges to a definite integral that gives the exact volume V,
