Nice. Some minor typos in the detail, though: "x2+y2=r", one "dO" that should not be there...
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this proof was one of my precious joys during undergraduate studies.
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that is, imagine looking your Gaussian from above...
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Oldie but a goodie.
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i) R^2=x^2+y^2 ii) Extra dθ
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Why always you have typos?
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May be more accessible to freshmen by finding volume of revolution (about the y-axis).
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x=R*cos(θ); y=R*sen(θ)
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Love this trick since 1st time I saw it. But, once my QM prof. asked: "Ok, now tell me what's PI have to do with probability"?
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