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See this video, but I can explain it without you watching it (nothing important is said and you can see the video in mute):

Basically all it does is cut the sphere into 'infinitesimal' pyramids with bases on the surface area of the sphere and main vertices aligning in the centre of the sphere.

The sum of the volumes of these pyramids simply turns out to be $\frac{4}{3}\pi r^3$. But this example shows that such methods don't necessarily work. Why does it work in this case?

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In the first example you can prove that the convergence is uniform. In the second you can prove it is not (the average distance between two corresponding points doesn't approach zero). So in the first one $\lim Volume(X) = Volume \lim X$, in the second you can't do that.

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