To evaluate the double integral $$\int\limits_0^8 {\left( {\int\limits_{y/2}^{\left( {y/2} \right) + 1} {\left( {{{2x - y} \over 2}} \right)dx} } \right)dy,\,\,} $$ we make the substitution $$u = \left( {{{2x - y} \over 2}} \right)$$ and $$v = {y \over 2}.$$ The integral will reduce to

The minimum value of the function $$f\left( x \right) = {x^3} - 3{x^2} - 24x + 100$$ in the interval $$\left[ { - 3,3} \right]$$ is

Consider a die with the property that the probability of a face with $$'n'$$ dots showing up is proportional to $$'n'.$$ The probability of the face with three dots showing up is _________.

Let $$X$$ be a random variable with probability density function $$f\left( x \right) = \left\{ {\matrix{ {0.2} & {for\,\left| x \right| \le 1} \cr {0.1} & {for\,1 < \left| x \right| \le 4} \cr 0 & {otherwise} \cr } } \right.$$

The probability $$P\left( {0.5 < x < 5} \right)$$ is _________.