The function f(x) = 8log_e x $$-$$ x² + 3 attains its minimum over the interval [1, e] at x = __________.

(Here log_e x is the natural logarithm of x.)

The value of the integral

$$\int\!\!\!\int\limits_D {3({x^2} + {y^2})dx\,dy} $$,

where D is the shaded triangular region shown in the diagram, is ___________ (rounded off to the nearest integer).

Let r = x² + y - z and z³ - xy + yz + y³ = 1. Assume that x and y are independent variables. At (x, y, z) = (2, -1, 1), the value (correct to two decimal places) of $${{\partial r} \over {\partial x}}$$ is ________________.

A three dimensional region $$R$$ of finite volume is described by $$\,\,{x^2} + {y^2} \le {z^3},\,\,\,0 \le z \le 1$$
Where $$x, y, z$$ are real. The volume of $$R$$ correct to two decimal places is __________.