The volume of the sphere v = 4π(x/2)3/3 =π×3/6.
ev=∫[a,b]πx^3/6(b-a)dx=[π(a+b)(a^2+b^2)]/24
Solution: probability density function of x: f(x)= 1/(b-a) x:[a, b]
F(x)=0 Other x
E(v)=∫(b，a) πx？ /6/(b-a) dx
= π/[6(b-a)] ∫(b，a) x？ Advanced (short for deluxe)
= π/[24(b-a)] x^4 |(b，a)
= π/[24(b-a)] (b^4 - a^4)
= π(a+b)(a？ +b？ )/24 ( 1)
That is, the mathematical expectation of sphere volume: E(v) = π(a+b)(a? +b？ )/24
Extended data:
In mathematics, the probability density function of a continuous random variable is a function that describes the possibility that the output value of this random variable is near a certain value point.
This refers to one-dimensional continuous random variables, and multi-dimensional continuous variables are similar.
Probability density function of random data: it represents the probability that the instantaneous amplitude falls within the specified range, so it is a function of amplitude. It varies with the range taken.
Baidu Encyclopedia-Density Function