<ol class="abc-list"><li><div class="list-item-paragraph">Differentiate \(\sqrt{1 - x^2}\) with respect to \(x\).<br>Hence evaluate \(\int_{-1}^1 \cos^{-1} x \, \mathrm{d}x\), giving your answer in exact form.<span class="inline-marks">[5]</span></div></li></ol>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">The finite region \(R\) is bounded by the curve \(y = \cos^{-1} (x - 3)\), where&nbsp;\(2 \leq x \leq 4\), the line \(y = \pi\) and the axes.</div><ol class="i-list"><li><div class="list-item-paragraph">Describe geometrical transformation which will map the graph of \(y = \cos^{-1} (x - 3)\) onto the graph of \(y = \cos^{-1} x\).<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph"> Hence or otherwise, find the area of \(R\) in exact form.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Find the exact value of the volume generated when&nbsp;\(R\) is rotated completely about the \(y\)-axis.</span><span class="inline-marks">[4]</span></div></li></ol>

Differentiate \(\sqrt{1 - x^2}\) with respect to \(x\).\n\nHence evaluate \(\displaystyle \int_{-1}^1 \cos^{-1} x \, \mathrm{d}x\), giving your answer in exact form.

Differentiate \(\sqrt{1 - x^2}\) with respect to \(x\).
Hence evaluate \(\displaystyle \int_{-1}^1 \cos^{-1} x \, \mathrm{d}x\), giving your answer in exact form.

geometrical transformation which will map the graph of \(y = \cos^{-1} (x - 3)\) onto the graph of \(y = \cos^{-1} x\).

The finite region \(R\) is bounded by the curve \(y = \cos^{-1} (x - 3)\), where \(2 \leq x \leq 4\), the line \(y = \pi\) and the axes.

Hence or otherwise, find the area of \(R\) in exact form.

Find the exact value of the volume generated when \(R\) is rotated completely about the \(y\)-axis.

2025 TJC P1 Q10

2025 TJC P1 Q10