# Math Help - Surface Area Integral

1. ## Surface Area Integral

Q: The curve $C$ has equation $y=\frac{2}{3}x^\frac{3}{2}$. The arc of $C$ from $(0, 0)$ to $(1, \frac{2}{3})$ is rotated through $2 \pi$ about the x-axis. The surface area generated has area $S$.
• Show that $S = \frac{4 \pi}{3}\displaystyle\int^1_0 x^\frac{3}{2} \sqrt{1+x} \, \mathrm{d}x$.
• Use the trapezium rule with $5$ equally spaced ordinates to find the approximate value for $S$, giving your answer to $2$ decimal places.
__________________
I know the formula is $S_x = 2 \pi \int y \ \mathrm{d}s$ however it is to be integrated in terms of $\mathrm{d}s$. Can someone help with this question. Thanks in advance.

2. Originally Posted by Air
Q: The curve $C$ has equation $y=\frac{2}{3}x^\frac{3}{2}$. The arc of $C$ from $(0, 0)$ to $(1, \frac{2}{3})$ is rotated through $2 \pi$ about the x-axis. The surface area generated has area $S$.
• Show that $S = \frac{4 \pi}{3}\displaystyle\int^1_0 x^\frac{3}{2} \sqrt{1+x} \, \mathrm{d}x$.
• Use the trapezium rule with $5$ equally spaced ordinates to find the approximate value for $S$, giving your answer to $2$ decimal places.
__________________
I know the formula is $S_x = 2 \pi \int y \ \mathrm{d}s$ however it is to be integrated in terms of $\mathrm{d}s$. Can someone help with this question. Thanks in advance.
$ds = \left ( \sqrt{1 + \frac{dy}{dx} ^2} \right ) dx$

3. Originally Posted by Air
• Use the trapezium rule with $5$ equally spaced ordinates to find the approximate value for $S$, giving your answer to $2$ decimal places.
^ How about this part? How can this be done? What does it mean by 'equally spaced ordinates'?

4. Originally Posted by Air
^ How about this part? How can this be done? What does it mean by 'equally spaced ordinates'?
Equally-spaced ordinates means the x-values you are taking for the function to approximate its area are spaced equally. In this case, that means your ordinates will be 0, 0.25, 0.5, 0.75 and 1. So the curve is $f(x)$. Your approximation will be $\frac{1}{2} \cdot \frac{1}{4}(f(0) + 2f(0.25) + 2f(0.5) + 2f(0.75) + f(1))$.