Integral Inequality involving the Euclidian Norm

Question

I have spent several hours trying to establish the inequality shown in the attached photo. Here we assume that $\vec{r}(t)$ is a vector function in $R^n$, and is integrable on $[a,b]$. I am in need of a little bit of initial guidance because I have run into a brick wall by trying to use absolute values and properties we already know about square roots and integrals.

Answer 1

I understand that $f_1,\dots, f_n$ are the components of the vector $\vec r(t)$. Since integration is componentwise, the vector $\int_a^b \vec r(t)\,dt$ has components $\int_a^b f_i(t)\,dt$. By adding the squares of these components, and taking square root, we get the norm of $\int_a^b \vec r(t)\,dt$. This is what happens on the left.

Not much happens on the right: the integrand $\|\vec r(t)\|$ is being replaced by a formula for the norm of $\|\vec r(t)\|$, namely $\sqrt{\sum f_i(t)^2}$.