The 2 sides of the parabola are symmetrical. So, the distance travelled as the object’s decreased = the distance traveled as the object’s velocity increased.
The equation for distance vs time for an object accelerating from 0 m/s is shown below.
d = vi * t + ½ * a * t^2
However, the initial velocity = 0 m/s
d = 0 * t + ½ * a * t^2
d = ½ * a * t^2
In angular motion:
θ = ωi * t + ½ * α * t^2
ωi = 0
α = (110 ÷ 144)
θ = ½ * (110 ÷ 144) * 144^2
θ = ½ * 110 * 144 = +7920
Initially, the wheel was accelerated at 0.7639 rpm for 144 seconds to reach a velocity of 110 rev/min. The number of revolutions that the wheel made during the acceleration = 7920 revolutions.
Then, the steam is shut off, so the friction of the bearings and of the air caused the wheel to decelerate at 0.7639 rpm for 144 seconds to reach a velocity of 0 rev/min.
The number of revolutions that the wheel made during the deceleration must equal 7920 revolutions, because a parabola is symmetrical!!
The distance is the area between the parabola and the x-axis.
The area under the left half of a parabola = area under right half of a parabola.
I hope this helps you!