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Answer to Question #5480 in Integral Calculus for kafi

Question #5480
A right circular cone is cut in half .The diameter of the base is 2r and the perpendicular height is h, By considering the volume of one slice of the half cone, and then summing all such slices, show from first principles that the volume is 1/6pi r^2h
Expert's answer
Consider a slice of the cone, a distance w down from the vertex, of thickness δw

Looking at the cut-face of the cone, you can see that the radius of the slice (s) can be obtained using the idea of similar triangles:
<img style="width: 155px; height: 39px;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPYAAAA+CAIAAACXy3FGAAADs0lEQVR4nO2c2YHrIAxFp6fphmZohVZciivxfHgHvCAkjDT3/L28SYLIsdhs/UwAmObn6wYAIAsUB8aB4sA4UBwYB4oD40BxYBwoDowDxYFxoDgwDhSPGYP7nXFh/LoxoB4onmPwMNwMUDzDnMj98HU7AAflig9+GcePCozBHdPe9jc6M+Hgl/AwaeGB4MzW9ctbtn8X/xKkLD635tjci1d0irH0ffBL+9PgQDElzozBzV2/5pph/im214sgKT5fUYcvW67AwytjcFqtWPPF1noozgDJmdl556r6njYXj1L04H+dO406ig1PjE5+HECB4Mz8S9T2PE3x03psDM6FcZu/rm1TKng6w0ISZ6HcGaZVf7XiY3D7ymxfKahNetEiCIZzUe4M03KOuGm4ffvgjytg+pqgG06ZZcI0hY9SZ7hyC1Hx5ZL0fm/Z7EbQLfjFNEVzRN1Q6Azb4QT16OdiPax9SE/7dVdc8wKjC8qcYcstVYqfGmDiSDDTr9lji/efd/OuMfh/NToUOcOnEw7wRbnd9sI0vwlQXJh1EMiavA7U0FwQKN6AW5PjmzEAM1C8FWs6z5p8+5/v2G8ZO4DhQZvi29LvNX39xrcmV8xalrcm2/mk8LV3cgwUb83t5ipN8+whCfkIzkInH9GluBXuTU73j199Wt+ifUeB4qXXdopcGDTqI6oJ6vZso+jgg/64QBtY+pnc7VD8i6B4s/h0Xmv2tzPD0s/kbsdEpTUCc/GFwyS603T+CboU174SktpRST+kIplr7+QYKN4K6X3xjdr74TR3cg5diitF7nTz5kbrDufk3wDFhZG9RyV+gOPwlTB8AYqLInynYeYSwSZ5DBQXRf5+8czMGfn7BBQHxiEoHuUNA0OivYjU0KCkXqHi0dzSwPOa9iJSh/AT4ATFM0/fKc579iLShvRDv7Vz8Ush4mWQGmcyEfFVSQUp0nWAaxXPDjLnilJJfam+iSLirZIKEsTrANcpnrc3VzNNy9z2+npkqZIKYuTrANconj1am3KPWWnhKqLp4RAHkJGvA0xW/MYGlvvd2vMiIkXRKKFBHeCKyrRPzagqI9Wap4hQ2lCEFnWACYqXlMfXsQP3HBHubJKhRR3gYsUzOwmDP22fpPttfSv+ENH6JzCcnyZ1gAsVz9ajSdbDuwv9j++PEU2ThjBU0qYOcJnilw+ErI1Jhek89T1GNE1I4lJw1wHOgzsNgXGgODAOFAfGgeLAOFAcGAeKA+NAcWAcKA6MA8WBcf4AlrivQ4EYHwcAAAAASUVORK5CYII=" alt="">
The cross sectional area of the (semi circular) slice=
<img style="width: 45px; height: 43px;" src="data:image/png;base64,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" alt="">
so the volume may be approximated to that of a half cylinder:
<img style="width: 52px; height: 43px;" src="data:image/png;base64,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" alt="">
provided that δw is sufficiently small.

Substituting for s, the volume of the slice (v) is just
<img style="width: 127px; height: 48px;" src="data:image/png;base64,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" alt="">
The volume (V) of the whole half cone is just the sum of the volumes of all the slices between w = 0 and w = h
<img style="width: 152px; height: 57px;" src="data:image/png;base64,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" alt="">
In the limiting case { as δw tends to 0 } , and taking out the constants π,r²,h²,2 this boils down to
<img style="width: 564px; height: 55px;" 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alt="">

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