Calculating the surface area of a sphere using dA

[jedishrfu]

Perhaps this bideo from 3blue1brown will give you a different perspective on the sphere

 

[PeterDonis]

To be a full circle

When you go from the South Pole to the North Pole, you only go through half a circle, not a full circle. That is what the integral we are talking about is doing. It is adding up the areas of infinitesimal circular "slices" of the Earth's surface, each "slice" taken at a different latitude, over the full range of latitudes from the South Pole to the North Pole. The area of each infinitesimal slice is the circumference of the circle at latitude $\theta$, which is $2 \pi r \cos \theta$, times the infinitesimal width of the slice, which is $r d \theta$. So we are integrating $2 \pi r^2 \cos \theta d \theta$ from $\theta = - \pi / 2$, the South Pole, to $\theta = \pi / 2$, the North Pole.

 

[anuttarasammyak]

To be a full circle, doesn't it have to be 2pi?

If you wonder about polar coordinates that includes a modified verion of measuring $\theta$ from xy plane not from usual z axis, you can see many explanations in school texts and web contents. Peter also has done a good explanation in the previous post.

If you question about coordinates system that I said in post #23 the figure of which you quoted, imagine as follows.
Imagine all the curves mentioning constant longitude spanning between the North pole and the South pole in your mind.
Imagine the equator circle is made of sponge wet with red ink. Turn it gradually along the East pole - the West pole axis. You see those curves are getting painted red left after the equator circle passes. How much angle should the equator circle turn to paint all parts of those curves ?
And how about it when not the full equator circle but only a half equator circle between EW axis sponge is wet ?
Colored red points on the globe surface have been shown already appointed by the coordinates.

 

[annamal]

If you wonder about polar coordinates that includes a modified verion of measuring $\theta$ from xy plane not from usual z axis, you can see many explanations in school texts and web contents. Peter also has done a good explanation in the previous post.

If you question about coordinates system that I said in post #23 the figure of which you quoted, imagine as follows.
Imagine all the curves mentioning constant longitude spanning between the North pole and the South pole in your mind.
Imagine the equator circle is made of sponge wet with red ink. Turn it gradually along the East pole - the West pole axis. You see those curves are getting painted red left after the equator circle passes. How much angle should the equator circle turn to paint all parts of those curves ?
And how about it when not the full equator circle but only a half equator circle between EW axis sponge is wet ?
Colored red points on the globe surface have been shown already appointed by the coordinates.

I get it now. Thank you!