# Thread: Areas of surfaces.

1. ## Areas of surfaces.

Here's my question:

Find the area of the surface created by the complete curve r=a when it is completely revolved about the x axis.Do not assume the formula for the surface area of a sphere.

Just by looking at it I did:

Which was incorrect.

So then I did it the way I think I was supposed to:

and this answer was also wrong. The book gave the answer as:

What's going wrong??

2. Originally Posted by Showcase_22
Here's my question:

Find the area of the surface created by the complete curve r=a when it is completely revolved about the x axis.Do not assume the formula for the surface area of a sphere.

Just by looking at it I did:

Mr F says: *Ahem* You're rotating a circle around the x-axis. This gives a sphere. You do recall that the surface area of a sphere is $\displaystyle {\color{red}4 \pi r^2}$, don't you?

Which was incorrect. Mr F says: Obviously.

So then I did it the way I think I was supposed to:

and this answer was also wrong. The book gave the answer as:

What's going wrong??
The formula is $\displaystyle S = 2 \pi \int_{\alpha}^{\beta} r \sin \theta \sqrt{ r^2 + \left( \frac{dr}{d\theta} \right)^2 } \, d \theta$.

So for your problem $\displaystyle S = 2 \times 2 \pi \int_{0}^{\pi/2} a \sin \theta \sqrt{ a^2 + 0^2 } \, d \theta = {\color{red}4} \pi \int_{0}^{\pi/2} a^2 \sin \theta \, d \theta$.

That's why your answer is wrong by a factor of 2.

3. I wasn't visualising it correctly. I was thinking of it in 2D form and not 3D.

When it's a 2D circle and it's rotated around the x axis, it only needs to be rotated by pi radians and not 2pi.

Thanks Mr Fantastic!