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Notes -
I appreciate putting someone in their place with the power of calculus, but a simpler demonstration would be a diagram of light hitting the earth. The portion of light captured by earth will be a circle, not the surface area of a half sphere, let alone the surface area of a full sphere.
The last (which is @Sky is assuming) would be attained if, with a paper earth, we cut earth open and lay it flat on the table to capture sunlight on all its surface.
Those are really fancy symbols there @popocatepetl
Are you sure you know what they mean?
I’ve got a strange hunch that you do not. The jig is up.
phrase of jig INFORMAL•NORTH AMERICAN
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Oh sure, that's the simplifying assumption which is presumed to be understood. Unfortunately, OP did not understand it. He may be confused by how the locally-valid rule I = Isuncos θincidence can give rise to the much simpler rule PEarth = Isunπr² when summed over all localities. The way to resolve such confusion is to evaluate the integral yourself and see that it all cancels out.
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