A2 June 2019 Paper 2 Q5

AQACurrent spec4 marksHyperbolic Functions Integration

5 A curve has equation \(y = \cosh x\)

Show that the arc length of the curve from \(x = a\) to \(x = b\), where \(0 \lt a \lt b\), is equal to

\[\sinh b - \sinh a\] [4 marks]

Mark scheme

Scheme	Marks	AO
Correctly substitutes the derivative of \(\cosh x\) in the formula for arc length. Condone missing limits.	M1	1.2
Correctly uses the hyperbolic identity \(1 + \sinh^2 x = \cosh^2 x\) to simplify the integrand. Condone \(\sqrt{1 + \sinh^2 x} = \cosh x\)	M1	1.1a
Correctly integrates \(\cosh x\)	M1	1.1a
Substitutes limits and uses a rigorous argument to show the required result	R1	2.1
	(4 marks)

Typical solution

\[s = \int_a^b \left(1 + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2\right)^{\frac{1}{2}}\mathrm{d}x\]\[\frac{\mathrm{d}y}{\mathrm{d}x} = \sinh x\]\[1 + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 = 1 + \sinh^2 x = \cosh^2 x\]\[s = \int_a^b (\cosh^2 x)^{\frac{1}{2}}\mathrm{d}x = \int_a^b \cosh x\,\mathrm{d}x\]\[= [\sinh x]_a^b\]\[= \sinh b - \sinh a\]