Define a function f(v)={c:v€[0,v(subscript)0)
0 : v€ [v(subscript)0,+∞}
On v €[0,+∞). Here and v(subscript)0 are positive constants.
A) Compute
∫v=0,+∞; f(v)dv
B) Suppose c=1/v(subscript)0, find
∫v=0,+∞;v f (v)dv
To compute the definite integral of a function, we need to find the antiderivative of the function and evaluate it at the upper and lower limits of integration.
A) Computing ∫v=0,+∞; f(v)dv:
1. For v € [0, v₀), the function f(v) is equal to c.
Therefore, the integral over this interval is ∫v=0,v₀ c dv.
Since c is a constant, we can take it outside the integral sign:
∫v=0,v₀ c dv = c ∫v=0,v₀ dv.
2. For v € [v₀, +∞), the function f(v) is equal to 0.
Therefore, the integral over this interval is ∫v=v₀,+∞ 0 dv.
Since the integrand is 0, this integral evaluates to 0.
Thus, the total integral of f(v) over the range v € [0,+∞) is the sum of the integrals over the intervals [0, v₀) and [v₀, +∞):
∫v=0,+∞; f(v)dv = c ∫v=0,v₀ dv + 0 = c [v] evaluated from 0 to v₀
= c (v₀ - 0)
= c v₀
= c v(subscript)0
Therefore, ∫v=0,+∞; f(v)dv = c v₀.
B) Suppose c = 1/v₀. We need to compute ∫v=0,+∞; v f(v)dv.
1. For v € [0, v₀), the function f(v) is equal to c = 1/v₀.
So, the integrand v f(v) = v * (1/v₀) = v/v₀.
The integral of v/v₀ over this interval is ∫v=0,v₀ v/v₀ dv.
As v/v₀ is a constant, we can simplify the integral:
∫v=0,v₀ v/v₀ dv = (1/v₀) ∫v=0,v₀ v dv.
2. For v € [v₀, +∞), the function f(v) is equal to 0.
Therefore, the integral over this interval is ∫v=v₀,+∞ v * 0 dv, which is 0.
Thus, the total integral of v f(v) over the range v € [0,+∞) is the sum of the integrals over the intervals [0, v₀) and [v₀, +∞):
∫v=0,+∞; v f(v)dv = (1/v₀) ∫v=0,v₀ v dv + 0 = (1/v₀) [v^2/2] evaluated from 0 to v₀
= (1/v₀) (v₀^2/2 - 0^2/2)
= (1/v₀) (v₀^2/2)
= v₀/2.
Therefore, ∫v=0,+∞; v f(v)dv = v₀/2.