Applications using double integrals: A lamina occupies the part of the disk x^2 + y^2 <= 1 in the first quadrant. Use polar coordinates to find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.

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Applications of integrals
asked 2021-02-03
Applications using double integrals:
A lamina occupies the part of the disk \(\displaystyle{x}^{{2}}+{y}^{{2}}\le{1}\) in the first quadrant. Use polar coordinates to find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.

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2021-02-04
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b) The systems \(\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ +\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}{\quad\text{and}\quad}{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ -\ {y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ -\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}\) are almost linear.
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Part B.) Find the vertical displacement D(t) between the raised hands of the two players for the time period after Boris has jumped (\(\displaystyle{t}{>}{t}_{{R}}\)) but before Arabella has landed. (Express youranswer in terms of t,\(\displaystyle{t}_{{R}}\), g,and H)
Part C.) What advice would you give Arabella To minimize the chance of her shot being blocked?
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