The region $A$ is bounded by the curves $x^2 + y^2 = 5$ and $y = x^2 + 1$ for \(y \geq 1\

Question

The region $A$ is bounded by the curves $x^2 + y^2 = 5$ and $y = x^2 + 1$ for $y \geq 1$. Find the volume when $A$ is rotated $\pi$ radians about the $y$-axis.[2] The region $B$ is bounded by the curve $y = \frac{2}{\sqrt{x^2 - 4x + 7}}$, the line $y = x - 2$, the line $x = 1$ and the $x$-axis. Find the exact volume when $B$ is rotated $2\pi$ radians about the $x$-axis. Give your answer in the form $a\sqrt{3}\pi^2 + b\pi$, where $a$ and $b$ are constants to be determined.[6]

Accepted Answer

(a) $\frac{\pi(20\sqrt5-41)}6=1.95\text{ units}^3$ to $3$ s.f. (b) $V=\frac{4\sqrt3}{9}\pi^2-\frac13\pi\text{ units}^3$, so $a=\frac49$ and $b=-\frac13$

2025 RVHS P2 Q3

2025 RVHS P2 Q3

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