Let $C$ be a semicircle of radius $r>0$ centered at the origin.
Let $P$ be a point on the x-axis whose coordinates are $P=(r+rt,0)$ where $t>0$.
Let $L$ be a line through $P$ which is tangent to the semicircle.
Let $A$ denote the triangular region between the circle and the line and above the x-axis (see figure.)

( Click on image for a larger view )

Find the exact area of $A$ in terms of $r$ and $t$.

$\mathrm{Area} (A) =$ .

Use a Maclaurin Polynomial to get a simple approximation for the area of $A$ for small $t$.

$\mbox{Area} (A) \approx$ .