The region specified by
$$\left\{ {\left( {\rho ,\varphi ,{\rm Z}} \right):3 \le \rho \le 5,\,\,{\pi \over 8} \le \phi \le {\pi \over 4},\,\,3 \le z \le 4.5} \right\}$$ in cylindrical coordinates has volume of ___________.

The value of the integral $$\int_{ - \infty }^\infty {12\,\,\cos \left( {2\pi t} \right){{\sin \left( {4\pi t} \right)} \over {4\pi t}}} dt\,\,$$ is __________.

The maximum area (in square units) of a rectangle whose vertices lie on the ellipse $${x^2} + 4{y^2} = 1\,\,$$ is

Which one of the following graphs describes the function? $$f\left( x \right) = {e^{ - x}}\left( {{x^2} + x + 1} \right)\,?$$