CAIE A-Level Further Math A2 2.4.4 Arc Length and Surface Area Questions

Practise calculating exact arc lengths and surfaces of revolution for curves with Further Mathematics Paper 2 questions and mark schemes.

Exam points

Calculate exact arc lengths and surfaces of revolution for curves.

[Maximum number: 7]

It is given that the arc length of C from x=a to x=2 a is $\ln 4$ , where a is a positive constant.

Show that $\cosh a=2$ and find, in logarithmic form, the exact value of a.

$\int_{a}^{2 a} \sqrt{1+\operatorname{cosech}^{2} x} \mathrm{~d} x$
Forms correct integral.

$\int_{a}^{2 a} \sqrt{\operatorname{coth}^{2} x} \mathrm{~d} x=\int_{a}^{2 a} \operatorname{coth} x \mathrm{~d} x$
Uses $\operatorname{coth}^{2} x-\operatorname{cosech}^{2} x=1$ .

$=[\ln \sinh x]_{a}^{2 a}=\ln \sinh 2 a-\ln \sinh a$
Integrates and substitutes limits.

$=\ln \frac{\sinh 2 a}{\sinh a}=\ln (2 \cosh a)$
Combines logarithms and uses double angle formula.

$\ln (2 \cosh a)=\ln 4 \Rightarrow \cosh a=2$
AG

$a=\ln \left(2+\sqrt{2^{2}-1}\right)=\ln (2+\sqrt{ } 3)$
Must reject $\ln (2-\sqrt{3})$ .

7

All question bank results loaded

CAIE A-Level Further Math A2 2.4.4 Arc Length and Surface Area Questions question 1