Q: The curve $\displaystyle C$ has equation $\displaystyle y=\frac{2}{3}x^\frac{3}{2}$. The arc of $\displaystyle C$ from $\displaystyle (0, 0)$ to $\displaystyle (1, \frac{2}{3})$ is rotated through $\displaystyle 2 \pi$ about the x-axis. The surface area generated has area $\displaystyle S$.

- Show that $\displaystyle S = \frac{4 \pi}{3}\displaystyle\int^1_0 x^\frac{3}{2} \sqrt{1+x} \, \mathrm{d}x$.
- Use the trapezium rule with $\displaystyle 5$ equally spaced ordinates to find the approximate value for $\displaystyle S$, giving your answer to $\displaystyle 2$ decimal places.

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I know the formula is $\displaystyle S_x = 2 \pi \int y \ \mathrm{d}s$ however it is to be integrated in terms of $\displaystyle \mathrm{d}s$. Can someone help with this question. Thanks in advance.