3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-04_560_1059_146_406}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows the curve with equation \(y = \ln ( 2 + \cos x ) , 0 \leq x \leq \pi\).
- Complete the table below for points on the curve, giving the \(y\) values to 4 decimal places.
- Giving your answers to 3 decimal places, find estimates for the area of the region bounded by the curve and the coordinate axes using the trapezium rule with
- 1 strip,
- 2 strips,
- 4 strips.
- Making your reasoning clear, suggest a value to 2 decimal places for the actual area of the region bounded by the curve and the coordinate axes.
| \(x\) | 0 | \(\frac { \pi } { 4 }\) | \(\frac { \pi } { 2 }\) | \(\frac { 3 \pi } { 4 }\) | \(\pi\) |
| \(y\) | 1.0986 | | | | 0 |
- continued
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-06_563_983_146_379}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows the curve with parametric equations
$$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$
The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.