Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of locally summable non-negative functions on $\mathbb{R}$.
Suppose that $\int_{a}^{b}x^2\,f_n(x)\,dx \xrightarrow[n\to\infty]{}0$ for every $a,b\in\mathbb{R}$ ,
1) can I conclude that $\int_{a}^{b}f_n(x)\,dx \xrightarrow[n\to\infty]{}0$ for every $a,b\in\mathbb{R}$ ?
2) can I conclude that $f_n(x) \xrightarrow[n\to\infty]{}0$ for Lebesgue-almost every $x\in\mathbb{R}$ ?
I think the answer to 1) should be yes, I tried to prove it using a change of variable ($y=x^3$) but I didn't menage.
Edit:I added the hypothesis $f_n\geq0$ later.