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Answer:
<u>Volume = 1.535</u>
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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