Volume = 1.535

Step-by-step explanation:

The region R is bounded by the equations:

y = √sin⁻¹x

y = √(π/2)

y = √(π/3)

x = 0

R is revolved around the x-axis so we will need f(y) for finding out the volume. We need to make x the subject of the equation and then replace it with f(y).

f(x) = √sin⁻¹x

y = √sin⁻¹x

Squaring both sides we get:

y² = sin⁻¹x

x = sin (y²)

f(y) = sin (y²)

Using the Shell Method to find the volume of the solid when R is revolved around the x-axis:

The limits a and b are the equations y = √(π/2) and y = √(π/3) which bound the region R. So, a = √(π/2) and b = √(π/3).

V = 2π sin (y²) dy

Integrating sin (y²) dy, we get:

-cos(y²)/2y

So,

V = 2π [-cos(y²)/2y] with limits √(π/2) and √(π/3)

V = 2π [(-cos(√(π/2) ²)/2*√(π/2)] - [(-cos(√(π/3) ²)/2*√(π/3)]

V = 2π [(-cos(π/2)/ 2√(π/2)) - ((-cos(π/3)/ 2√(π/3))]

V = 2π [ 0 - (-0.5/2.0466)]

V = 2π (0.2443)

V = 1.53499 ≅ 1.535

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