Differentiate $\sqrt{1 - x^2}$ with respect to $x$. Hence evaluate \(\displaystyle\int\nolimits_

Question

Differentiate $\sqrt{1 - x^2}$ with respect to $x$. Hence evaluate $\displaystyle\int\nolimits_{-1}^1 \cos^{-1} x \, \mathrm{d}x$, giving your answer in exact form.[5] 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. Describe geometrical transformation which will map the graph of $y = \cos^{-1} (x - 3)$ onto the graph of $y = \cos^{-1} x$.[1] Hence or otherwise, find the area of $R$ in exact form.[2] Find the exact value of the volume generated when $R$ is rotated completely about the $y$-axis.[4]

Accepted Answer

(a)$-\frac{x}{\sqrt{1-x^2}}$, $\pi$ (b)(ii)$3\pi$ (c)$\frac{19}{2}\pi^2$

2025 TJC P1 Q10

2025 TJC P1 Q10