Real examples of non-positive definite integrals

Question

We need to get some examples of non-positive definite integrals in real practice (mathematical physics, engineering, mathematical statistics and etc).

Answer 1

A car travels along a straight line. At time $t$, the car is travelling with velocity $$ v\left(t\right)=\sin t. $$ Negative velocity means that it is travelling in the "left" direction (i.e. from $1$ to $0$), while positive velocity means that it is travelling in the "right" direction (i.e. from $0$ to $1$). Let $r\left(t\right)$ be the position of the car at time $t$. Find an expression for $r\left(t\right)$ given $r\left(0\right)=0$ (the car is initially at position 0). \begin{align*} r\left(t\right)-r\left(0\right)=r\left(t\right) & =\int_{0}^{t}v\left(s\right)ds\\ & =\int_{0}^{t}\sin sds\\ & =1-\cos t. \end{align*}

Notice that $\sin t$ takes on values from $-1$ to $+1$ (i.e. the integrand is not everywhere nonnegative), as per your request. Cars do not usually travel along lines, and the velocity function is strange, but less idealistic examples can similarly be produced.