The temperature at r distance from centre is

So when a small length ds is moved to from the centre, it will vary in proportion to the temperature T.

The new length

for some constant k.

But if you think about it, when you're at the centre w/ r = 0, you'll get

which is true iff

Therefore

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A more intuitive way to think of it is using the ratio of lengths (L).

We have a ratio

.

Because the lengths themselves are proportional to the specific temperature at the radius they are at,

where

are the distance from center the lengths dr and ds are at.

At

At

So we have

Yes. That's another way of saying this is the value of all the sums of dr. Because dr changes with distance from radius, we sum all the small values of dr at each distance to make the radius. If we make dr sufficiently small, as with integration, it will be a good estimation for the radius.