## Question 6:

### Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.5 \times 4 \times (4\sqrt{1} + 8\sqrt{5} + 4\sqrt{9})$ | M1* | Attempt $y$-values at $x = 1, 5, 9$ only. Must be using $y$, not integration. Allow slips e.g. $\sqrt{4x}$ as long as clearly intended as $y$. Allow decimal equiv for $y_1$ (8.94). Allow M1 for 4, 20, 72 (omitting $\sqrt{}$). M0 if other $y$-values found (unless not used in trap rule) |
| $= 2(16 + 8\sqrt{5})$ | M1d* | Attempt correct trapezium rule, inc $h = 4$. Correct structure including 'big brackets' seen or implied. Allow 2 used for $\frac{1}{2}h$. Allow slips when calculating $y$ values, but all other aspects must be correct |
| $= 32 + 16\sqrt{5}$ **AG** | A1 | Obtain $32 + 16\sqrt{5}$. Must come from exact working. isw if exact answer found first, then decimal equivalent stated |

### Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch showing correct graph of $y = 4\sqrt{x}$ and two trapezia | B1* | Correct graph shown, existing for at least $1 \leq x \leq 9$. Exactly two trapezia must be shown, of roughly equal widths, with top vertices on the curve |
| Curve is above tops of trapezia | B1d* | Must refer to tops of trapezia. Allow 'trapezium' rather than 'trapezia'. B0 for 'trapezia are below curve' (i.e. 'top' not used). Could shade gaps on diagram but some text also required. B0 for 'some area not calculated' unless clear which area. Concave/convex is B0. B1 for decreasing gradient (but B0 for decreasing curve). No sketch is B0 irrespective of explanation |

### Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_1^9 4x^{\frac{1}{2}}\,dx = \left[\frac{8}{3}x^{\frac{3}{2}}\right]_1^9$ | M1 | Obtain $kx^{\frac{3}{2}}$. Any numerical $k$, including 4. Any exact equiv for the index |
| $= 72 - \frac{8}{3}$ | A1 | Obtain $\frac{8}{3}x^{\frac{3}{2}}$. Allow unsimplified coefficient, inc $\frac{4}{1.5}$ or $\frac{2}{3} \times 4$. Allow non-exact decimal i.e. 2.7, 2.67 etc. Allow $+ c$ |
| $= 69\frac{1}{3}$ | M1 | Attempt correct use of limits. Must be $F(9) - F(1)$ i.e. subtraction with limits in correct order. Allow use in any function other than the original, including from differentiation. Allow processing errors e.g. $\left(\frac{8}{3} \times 9\right)^{1.5}$ |
| | A1 | Obtain $69\frac{1}{3}$, or any exact equiv. Allow improper fraction or recurring decimal. A0 for $69.333\ldots$ A0 for $69\frac{1}{3} + c$ |

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