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

If $y = f(x)$ is a monotonically increasing function such that $\left(\dfrac{dy}{dx}\right)^2 = 6 - \dfrac{dy}{dx}$ and $y(0) = 5$, then $y(3) = \cdots$