Question 45 – 11.5 Integration by Substitution - Questions

A precision-engineered acoustic horn is modeled as a solid of revolution. The internal profile of the horn is defined by the curve $C\,$ with equation

$\frac{2}{\sqrt{\sqrt{x} + x\sqrt{x}} \arctan(\sqrt{x})}$

for $13≤x≤1\frac{1}{3} \le x \le 1$ . The region $R$ is bounded by the curve $C$ , the $x$ -axis, and the vertical lines $\frac{1}{3}$ and $x = 1$ .

The internal volume $V\,$ of the horn is formed by rotating the region $R$ through $360^\circ\,$ about the $x$ -axis.

Using the substitution $tan⁡u=x\tan u = \sqrt{x}$ ,

show that the volume $V$ is given by

$\int_{a}^{b} \frac{1}{u^2} \, du$

where $k, a$ and $b$ are constants to be found.

Hence, using algebraic integration, find the exact value of $V$ .