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Math Help - Lower sum for partition

  1. #1
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    Lower sum for partition

    f:[0,pi]---R be defined by f(x)=\sqrt sinx

    by considering the lower sum for the partition P={0,pi/6,5pi/6,pi},prove that \int_0^{\pi}f\ge\sqrt2pi/3

    I_1=[0,\pi/6], d_1=\pi/6

    I_2=[\pi/6,\pi*5/6], d_2=\pi*2/3

    I_2=[\pi*5/6,\pi], d_3=\pi/6

    L(f,P)=\sum_{i=1}^3m_id_i How do i find m_i for each of the above interval.

    Thanks
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  2. #2
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    f is increasing on [0,\pi/2] and decreasing on [\pi/2,\pi]. On the former, pick the leftmost points. On the later, pick the rightmost points. It doesn't matter on the middle intrval, since its symmetric about the maximum.
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