Volume and surface area of revolution

A question is this type if and only if it applies a reduction formula to calculate volume of revolution, surface area of revolution, or arc length of a curve.

2 questions · Hard +2.0

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CAIE FP1 2017 November Q8

11 marks Challenging +1.8

Find the value of $I _ { 2 }$.
Show that, for $n > 2$, $$( n - 1 ) I _ { n } = 2 ^ { \frac { 1 } { 2 } n - 1 } + ( n - 2 ) I _ { n - 2 }$$
The curve $C$ has equation $y = \sec ^ { 3 } x$ for $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$. The region $R$ is bounded by $C$, the $x$-axis, the $y$-axis and the line $x = \frac { 1 } { 4 } \pi$. Find the volume of revolution generated when $R$ is rotated through $2 \pi$ radians about the $x$-axis.

Pre-U Pre-U 9795/1 2011 June Q11

15 marks Hard +2.3

Let $I_n = \int_0^{\frac{\pi}{2}} \sec^n t \, dt$ for positive integers $n$. Prove that, for $n \geqslant 2$, $$(n - 1)I_n = \frac{2^{n-2}}{(\sqrt{3})^{n-1}} + (n - 2)I_{n-2}.$$ [5]
The curve with parametric equations $x = \tan t$, $y = \frac{1}{4}\sec^2 t$, for $0 \leqslant t \leqslant \frac{1}{4}\pi$, is rotated through $2\pi$ radians about the $x$-axis to form a surface of revolution of area $S$. Show that $S = \pi I_5$ and evaluate $S$ exactly. [10]