Not that Snake!

Here is the general expression for $g(x)$ :

$\displaystyle g(x)=\int_{0}^{x}f(t)dt$

= $\displaystyle \int_{0}^{\lfloor x \rfloor}f(t)dt + \int_{\lfloor x \rfloor}^{x}f(t)dt$

= $\displaystyle \lfloor x \rfloor\int_{0}^{1}f(t)dt + \int_{0}^{\{x\}}f(t)dt$ (where, $f(x)=x-x^2$ )

= $\displaystyle \frac{\lfloor x \rfloor+\{x\}^2(3-2\{x\})}{6}$

Condition $f(x+1) = f(x)$ for all $x\in\mathbb{R}$ means the function $f$ is a periodic function with period 1. Since $f(x) = x^2 - x^3$ for $0\leq x \leq 1$ , then $g(6.3) = \int_{0}^{6.3}\:f(x)\:dx = 6 \int_{0}^{1}\:f(x)\:dx + \int_{0}^{0.3}\:f(x)\:dx = 1.036$