## Question 3:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(\frac{dx}{d\theta}\right) = 2\sinh 2\theta$ **and** $\left(\frac{dy}{d\theta}\right) = 4\cosh\theta$ | B1 | Or equivalent correct derivatives |
| $A = (2\pi)\int 4\sinh\theta\sqrt{"2\sinh 2\theta"^2 + "4\cosh\theta"^2}\,d\theta$ or $A = (2\pi)\int 4\sinh\theta\sqrt{1+\left(\frac{"4\cosh\theta"}{"\,2\sinh 2\theta"}\right)^2}\cdot 2\sinh 2\theta\,d\theta$ | M1 | Use of correct formula including replacing $dx$ with $"2\sinh 2\theta"\,d\theta$ if chain rule used; allow omission of $2\pi$ |
| $A = 32\pi\int\sinh\theta\cosh^2\theta\,d\theta$ or $A = 32\pi\int(\sinh\theta + \sinh^3\theta)\,d\theta$ | B1 | Completely correct expression for $A$ with square root removed; mark may be recovered later if $2\pi$ introduced later |
| $A = \frac{32\pi}{3}\left[\cosh^3\theta\right]_0^1$ | dM1, A1 | M1: valid attempt to integrate a correct expression or multiple of correct expression — dependent on first M1. A1: correct expression |
| $= \frac{32\pi}{3}\left[\cosh^3 1 - 1\right]$ | ddM1, A1 | M1: uses limits 0 and 1 correctly — dependent on **both** previous M's. A1: Cao and cso (no errors seen) |

**(7 marks)**

**Alternative Integration (last 4 marks):**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\sinh\theta\cosh^2\theta\,d\theta = \int\sinh\theta(1+\sinh^2\theta)\,d\theta = \int(\sinh\theta+\sinh^3\theta)\,d\theta$ | | |
| $\int\!\left(\sinh\theta + \frac{1}{4}\sinh 3\theta - \frac{3}{4}\sinh\theta\right)d\theta = \frac{1}{4}\int(\sinh\theta+\sinh 3\theta)\,d\theta$ | | |
| $= \frac{1}{4}\cosh\theta + \frac{1}{12}\cosh 3\theta$ | dM1, A1 | dM1: $\int\sinh\theta\cosh^2\theta\,d\theta = p\cosh\theta + q\cosh 3\theta$. A1: $32\pi\!\left[\frac{1}{4}\cosh\theta+\frac{1}{12}\cosh 3\theta\right]$ |
| $A = 8\pi\!\left[\cosh\theta + \frac{1}{3}\cosh 3\theta\right]_0^1 = 8\pi\!\left(\cosh 1 + \frac{1}{3}\cosh 3 - \cosh 0 - \frac{1}{3}\cosh 0\right)$ | ddM1, A1 | M1: uses limits 0 and 1 correctly — dependent on both previous M's. A1: Cao |
| $\frac{32\pi}{3}\left[\cosh^3 1 - 1\right]$ | | |

**Alternative Cartesian Approach:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = 1 + \frac{y^2}{8}$ | B1 | Any correct Cartesian equation |
| $\frac{dx}{dy} = \frac{y}{4}$ or $\frac{dy}{dx} = \frac{\sqrt{2}}{(x-1)^{\frac{1}{2}}}$ | B1 | Correct derivative |
| $A = \int 2\pi\cdot y\sqrt{1+\left(\frac{y}{4}\right)^2}\,dy$ or $A = \int 2\pi\cdot\sqrt{8(x-1)}^{\frac{1}{2}}\sqrt{1+\left(\frac{2}{x-1}\right)}\,dx$ | M1 | Use of correct formula |
| $A = 2\pi\times\frac{2}{3}\times 8\left(1+\frac{y^2}{16}\right)^{\frac{3}{2}}$ or $\frac{4\pi\sqrt{8}}{3}\left(x+1\right)^{\frac{3}{2}}$ | dM1, A1 | M1: convincing attempt to integrate — dependent on first M1 (allow omission of $2\pi$). A1: completely correct expression for $A$ |
| $A = 2\pi\times\frac{2}{3}\times 8\left(1+\sinh^2 1\right)^{\frac{3}{2}} - 2\pi\times\frac{2}{3}\times 8$ or $2\pi\times\frac{2}{3}\times\sqrt{8}\left(1+\cosh 2\right)^{\frac{3}{2}} - \frac{32\pi}{3}$ | ddM1 | Correct use of limits ($0\to 4\sinh 1$ for $y$ or $1\to\cosh 2$ for $x$) |
| Use $1+\sinh^2 1=\cosh^2 1$ to give $\frac{32\pi}{3}\left[\cosh^3 1-1\right]$; or use $\cosh 2=2\cosh^2 1-1$ to give $\frac{32\pi}{3}\left[\cosh^3 1-1\right]$ | A1 | |

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