Q. Let $f:[1,2] \rightarrow \mathbb{R}$ be a non negative and integrable function with $\int_{1}^{2} f(x) d x=\frac{2}{3} .$ Define $g:[1, \sqrt{2}] \rightarrow \mathbb{R}$ by $g(x)=f\left(x^{2}\right)$, then which of the following are necessarily true?
(A) $g$ is integrable on $[1, \sqrt{2}]$
(B) $\int_{1}^{\sqrt{2}} g(x) d x=0$
(C) $\frac{\sqrt{2}}{6} \leq \int_{1}^{\sqrt{2}} g(x) d x \leq \frac{1}{3}$
By the Riemann integrability of composite functions, option A is correct, but how to find a range for $\int_{1}^{\sqrt{2}} g(x) d x$? I feel that I have to take a particular Riemann sum.