7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{33292670-3ad0-4125-a3bb-e4b7b21ed5f4-22_678_776_248_639}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve \(C\) with equation
$$r = 1 + \tan \theta \quad 0 \leqslant \theta < \frac { \pi } { 3 }$$
Figure 1 also shows the tangent to \(C\) at the point \(A\).
This tangent is perpendicular to the initial line.
- Use differentiation to prove that the polar coordinates of \(A\) are \(\left( 2 , \frac { \pi } { 4 } \right)\)
The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the tangent at \(A\) and the initial line.
- Use calculus to show that the exact area of \(R\) is \(\frac { 1 } { 2 } ( 1 - \ln 2 )\)