The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is

A parametric curve defined by $$x = \cos \left( {{{\pi u} \over 2}} \right),y = \sin \left( {{{\pi u} \over 2}} \right)\,\,$$ in the range $$0 \le u \le 1$$ is rotated about the $$x-$$axis by $$360$$ degrees. Area of the surface generated is

A six-face fair dice is rolled a large number of times. The mean value of the outcomes is _________.

$$\,\,P\,\,\,\left( {0,3} \right),\,\,Q\,\,\,\left( {0.5,4} \right),\,\,$$ and $$\,\,R\,\,\,\left( {1,5} \right)\,\,\,$$ are three points on the curve defined by $$\,\,f\left( x \right),\,\,$$ Numerical integration is carried out using both Trapezoidal rule and Simpson's rule within limits $$x=0$$ and $$x=1$$ for the curve. The difference between the two results will be