FP3 June 2012 Q2
2.
The curve \(C\), shown in Figure 1, has equation \[y = \frac{1}{3}\cosh 3x, \qquad 0 \leqslant x \leqslant \ln a\]
where \(a\) is a constant and \(a > 1\)
Using calculus, show that the length of curve \(C\) is \[k\left(a^3 - \frac{1}{a^3}\right)\]
and state the value of the constant \(k\). (6)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \sinh 3x\) | B1 |
| so \(s = \displaystyle\int \sqrt{1 + \sinh^2 3x}\,\mathrm{d}x\) | M1 |
| \(\therefore s = \displaystyle\int \cosh 3x\,\mathrm{d}x\) | A1 |
| \(= \left[\tfrac{1}{3}\sinh 3x\right]_0^{\ln a}\) | M1 |
| \(= \tfrac{1}{3}\sinh 3\ln a = \tfrac{1}{6}\left[e^{3\ln a} - e^{-3\ln a}\right]\) | DM1 |
| \(= \tfrac{1}{6}\left(a^3 - \dfrac{1}{a^3}\right)\) (so \(k = 1/6\)) | A1 |
| (6 marks) |
Notes
1B1: cao
1M1: Use of arc length formula, need both \(\sqrt{\ }\) and \(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2\).
1A1: \(\displaystyle\int \cosh 3x\,\mathrm{d}x\) cao
2M1: Attempt to integrate, getting a hyperbolic function o.e.
3M1: depends on previous M mark. Correct use of ln a and 0 as limits. Must see some exponentials.
2A1: cao