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Math Help - volume of hemisphere using Rieman's sums

  1. #1
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    volume of hemisphere using Rieman's sums

    We have started getting volumes and areas using Reiman's sums and then turning them into definite integrals. I get some of this but there is one part when we take the antiderv that is not getting through to me!!!


    If we have a hemsishere that has a radius of 7 cm^3 then the volume of a slice would be pi r^2 dh of slice


    so our integral would be from 0 to 7 of pi(7^2 -h^2) dh

    when this has its antiderv taken the book says it is


    pi(7^2h - 1/3h^3 ) from 0 to 7 and then they say this is equal to

    2/3 pi7^3

    Please tell me how they get this since I thought the h was so small we do not use it and the other term would just equal to pi 7^2 (7)

    In other words how do they get the 2/3?????????

    THanks for any help you math minded folks can provide!
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  2. #2
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    Are you using washers?. We can generate the volume by revolving the upper half circle about the x-axis.

    f(x)=y=\sqrt{r^{2}-x^{2}}

    V={\pi}\int_{-r}^{r}(r^{2}-x^{2})dx

    {\pi}\left[r^{2}x-\frac{x^{3}}{3}\right]=\frac{4}{3}{\pi}r^{3}

    For a hemisphere, divide by 2. Just sub in your 7 to find the volume when r=7

    The reason they got the 2/3 is because you want half of the sphere and 1/2 of 4/3 is 2/3.
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