Question #40842

Let $C$ be a curve parameterized by $x(t) = 3\sec t$ and $y(t) = 2\tan t$ for $t \in \left[0, \frac{\pi}{4}\right]$. A family of lines $L_t$ is given by the equation $\begin{vmatrix} X & Y & 1 \\ x(t) & y(t) & 1 \\ x'(t) & y'(t) & 0 \end{vmatrix} = 0$ Let $\Delta(t)$ denote the area of the region bounded by the line $L_t$ and the pair of straight lines $4X^2 - 9Y^2 = 0$. If $D(t) = \begin{vmatrix} x(t) & y(t) & \Delta(t) \\ x'(t) & y'(t) & \Delta'(t) \\ x''(t) & y''(t) & \Delta''(t) \end{vmatrix}$, evaluate the integral $I = \int_{0}^{\pi/4} D(t) dt$.