So why didn't our construction work? Because we didn't integrate along our slices. Neither of our integration variables are constant along the lines shown! As a rule of thumb, this doesn't matter for the first integration, because conversion to a single variable for the final integration effectively compensates. But it does matter for the second. A better choice of variables for this slicing would therefore have been x and z, and it is easy to verify that this does lead to the correct answer. But there is also another way.
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