Thread: Surface area of a sphere

1. Surface area of a sphere

How is the formula $4\pi r^2$derived?

My teacher gave me this explaination, but I didn't get a chance to ask him questions.

$x^2+y^2=r^2$

$y=\sqrt{r^2-x^2}$

Taking the derivative of y

$\int^r_0 2\pi y[1+(\frac {dy} {dx})^2]$

further integration (not typing out the steps)

$2\pi r^2$

Where do you get $2\pi$ from? Why do you need to find the length of the curve?

2. 2pi radians is a full turn round the axis and the length of the curve is what you are multiplying it by. It maps out the surface area. (I think).

3. Originally Posted by chengbin
How is the formula $4\pi r^2$derived?

My teacher gave me this explaination, but I didn't get a chance to ask him questions.

$x^2+y^2=r^2$

$y=\sqrt{r^2-x^2}$

Taking the derivative of y

$\int^r_0 2\pi y[1+(\frac {dy} {dx})^2]$

further integration (not typing out the steps)

$2\pi r^2$

Where do you get $2\pi$ from? Why do you need to find the length of the curve?
go to the link ... good explanation + examples

Arc Length and Surface Area

4. Originally Posted by sean.1986
2pi radians is a full turn round the axis and the length of the curve is what you are multiplying it by. It maps out the surface area. (I think).
My teacher only did half of a sphere (since you can just multiply it by 2 to get a whole). Wouldn't $\pi$ be the full turn around the axis?

5. You might be right. It's been a while since I did stuff like that.

EDIT: Just read it's the circumference (2 pi r) times the diameter (2r), which would make sense, because if you rotated a circle 180 degrees about the center, the point at either side would travel the length of the circumference.

6. Originally Posted by sean.1986
You might be right. It's been a while since I did stuff like that.
Then how would that work? $2\pi$ works, but I don't see how.

It is pretty good that you guys still remember. In fact I'm shocked on how much you guys know and remember. I did optimization and Rolle's Theorem a few months ago and now I forgot a lot of it. I have to review them now.

Are you supposed to "understand" calculus? I know how to do calculus, but I don't "understand" it. For example, I don't see how a trigonometric function can substitute for x in certain types of square root integration.