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Math Help - two Q's: finding center of mass of a bounded region, and finding length of curve

  1. #1
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    two Q's: finding center of mass of a bounded region, and finding length of curve

    I'm working on two problems. I've some rather rough solutions worked out, but am pretty sure they are wrong. Please show me step-wise the correct way to solve the following problems:

    1.) find the center of mass of a region bounded by y=x and y=1-2x^2.

    2.) sketch and calculate the length of the polar curve y=theta^2 from theta=0 to theta=4pi.
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  2. #2
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    2.) sketch and calculate the length of the polar curve y=theta^2 from theta=0 to theta=4pi.
    Shouldn't that be r={\theta}^{2}?.

    Anyway, arc length in polar is given by L=\int_{\alpha}^{\beta}\sqrt{r^{2}+\left(\frac{dr}  {d{\theta}}\right)^{2}}d{\theta}

    \int_{0}^{4\pi}\sqrt{{\theta}^{4}+4{\theta}^{2}}d{  \theta}

    \int_{0}^{4\pi}\sqrt{{\theta}^{2}({\theta}^{2}+4)}  d{\theta}

    =\int_{0}^{4\pi}{\theta}\sqrt{{\theta}^{2}+4}d{\th  eta}

    You could use trig sub to solve this by using the sub:

    {\theta}=2tan(x), \;\ d{\theta}=2sec^{2}(x)dx
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