CAIE A-Level Further Math 2.4.4 Arc length & surface area

Practise setting up arc length and surface area integrals from Cartesian, parametric or polar curves, often giving exact answers.

Exam points

differentiate parametric or polar equations and simplify the arc length integrand
set up a surface of revolution integral about an axis and integrate it exactly

[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.

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

M1

Forms correct integral.

$\int_{a}^{2 a} \sqrt{\operatorname{coth}^{2} x} \mathrm{~d} x=\int_{a}^{2 a} \operatorname{coth} x \mathrm{~d} x$

M1 A1

Uses $\operatorname{coth}^{2} x-\operatorname{cosech}^{2} x=1$ .

$=[\ln \sinh x]_{a}^{2 a}=\ln \sinh 2 a-\ln \sinh a$

M1

Integrates and substitutes limits.

$=\ln \frac{\sinh 2 a}{\sinh a}=\ln (2 \cosh a)$

M1

Combines logarithms and uses double angle formula.

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

A1

AG

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

A1

Must reject $\ln (2-\sqrt{3})$ .

7

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2.4.4—Arc length and surface area question 1