Surface area of a sphere

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Question from maria, a student:

I've read your solution to the problem of how to derive the formula of the surface area of the sphere.
But i've got a question that initially you demenstrate the area is apprximately equal to that of the lateral area of the cylinder. Afterwards you simply say that the two areas are exactly equal. I'm very confused, how you connect your two statements together. Can you explain to me how they are exactly equal? Your quick reply would be appreciated. Thanks for your help!

Hi there.

If you take a finite strip of the cylinder and sphere, they are approximately equal - I don't think you are questioning this. The leap to exactness is due to taking a smaller and smaller strip height h_c and h_s. The smaller these are, the more exact the comparison becomes. When you use calculus, you make this height infinitely small, so the distortion (comparison of approximation to exact value) becomes infinitely small.

It's a bit like comparing 0.9 to 1. That's approximately the same, but if we make it closer, then 0.99 is approximately 1, but certainly nearer than the first one.

Similarly, 0.99999....repeating forever is infinitely close to 1, so we say it is equal to 1. Thus it is exact.
(1/9) = 0.1111...., 9/9 = 0.9999.... = 1.

Stephen La Rocque.>

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