Find $\int _{[0,1]} f \, \, d \lambda$

Question

Let $e_k(x)$ be the $k$-th digit after the decimal point $x$ written as a decimal. Let $f(x) = e_2(x)$ (so if $f(0.521)=2$)

Find $$\int _{[0,1]} f \, \, d \lambda$$

I have no idea how to do this. Please help.

Answer 1

You have that $$\int_{[0,1]}fd\lambda=\int_{[0,0.1]}fd\lambda+\dots+\int_{[0.9,1]}fd\lambda.$$ Note that $f(x+0.1)=f(x)$ to get that $$\int_{[0,1]}fd\lambda=10\times\int_{[0,0.1]}fd\lambda.$$ Now write $$\int_{[0,0.1]}fd\lambda=\int_{[0,0.01]}fd\lambda+\dots+\int_{[0.09,0.1]}fd\lambda=0.01\times[1+\dots+9]$$ to finally get $$\int_{[0,1]}fd\lambda=10\times 0.01\times45=4.5.$$