Edexcel A-Level Mathematics A2 Fp3 4 6 The Questions

Practise finding arc lengths and surfaces of revolution from Cartesian or parametric curves, simplifying derivatives before exact integration.

Exam points

set up arc length using dy/dx or parametric derivatives, then simplify the integrand
form a surface of revolution integral about the x-axis using 2πy ds
evaluate exact lengths or areas involving e^t, hyperbolic functions or inverse trig

[Maximum number: 4]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Figure 1

Figure 1 shows the curve with equation

y=\ln\left(\tanh\frac{x}{2}\right) \quad 1\leqslant x\leqslant 2

Show that the length, s, of the curve is given by

s=\int_1^2 \coth x\,\mathrm{d}x

\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\tanh(x/2)}\cdot\frac12\operatorname{sech}^2(x/2)

=\frac{1}{2\sinh(x/2)\cosh(x/2)}=\operatorname{cosech}x

\sqrt{1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2}=\sqrt{1+\operatorname{cosech}^2x}

=\sqrt{\coth^2x}=\coth x\quad (1\le x\le2)

s=\int_1^2\coth x\,\mathrm{d}x

M1: obtains an appropriate expression for dy/dx.
M1: applies the arc length formula.
ddM1: uses identities to obtain 1+(dy/dx)^2 in terms of x.
A1*: reaches the printed integral with no errors.

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Edexcel A-Level Mathematics A2 Fp3 4 6 The Questions question 1