1.11 H: Integration

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Question 20

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.

[5]

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

[3]

1.11 H: Integration

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Question 20

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.

[5]

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

[3]

1.11 H: Integration

Question 20

1.11 H: Integration Questions

1.11 H: Integration

Question 20

1.11 H: Integration Questions