Variance of values of a function over an interval

Question

I want to find out the spread of values that a particular function can take over an interval. How do I calculate the variance of the values of the function over that interval? For example, if I want to calculate the mean value of the function over an interval, I find out the area under the curve and divide it by the width of the interval. I want to know if there is something similar for variance.

Answer 1

You have described the mean as $$\frac1{b-a}\int_a^b f(x)\, dx$$ Let's call that value $m_f$

Then you could describe the variance as $$\frac1{b-a}\int_a^b \left(f(x)-m_f\right)^2\, dx$$ which is equal to $$\left(\frac1{b-a}\int_a^b f(x)^2\, dx\right) - m_f^2$$