Edexcel IAL Mathematics FP3.4.6 arc length and surfaces of revolution

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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FP3.4.6 - The question 1