If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\int_0^2 x^2[x]\, dx$ is equal to

If $\int_0^1 \dfrac{dx}{\sqrt{x + 1} - \sqrt{x}} = \sqrt{2}k$, then the value of $k$ is...

The area of the region lying in the first quadrant and bounded by the curve $y = 4x^2$, the Y-axis, and the lines $y = 2$, $y = 4$ is...

The area bounded by the curves $y = |x| - 1$ and $y = -|x| + 1$ is