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1.  Integrate \(\int\int\int_Q x^2+y^2+z^2 dV\) where \(Q = \{(x,y): x^2+y^2+z^2 \leq 1, z^2 \geq x^2+y^2, z\geq0\}\)

a.  \(\frac{2\pi}{3}(1-\frac{\sqrt{2}}{2})\)

b.  \(\frac{2\pi}{5}(1-\frac{\sqrt{2}}{2})\)

c.  \(\frac{4\pi}{3}\)

d.  \(\frac{4\pi}{5}\)

e.  \(\frac{5\pi}{2}\)

2.  If we have a solid over the region given in problem 1, with the density function \(\rho(x,y,z) = x^2+y^2+z^2\), what can you say about its center of mass? (Do the integration if you must, but if you think about it, you should be able to avoid all that work.)

a.  It's at the origin.

b.  \(\overline{x}\) and \(\overline{y}\) are both 0, and \(0\leq\overline{z}\leq\frac{1}{2}\)

c.  \(\overline{x}\) and \(\overline{y}\) are both 0, and \(\frac{1}{2}\leq\overline{z}\leq 1\)

d.  \(\overline{x}\) and \(\overline{y}\) are both 0, and \(1\leq\overline{z}\)

e.  \(\overline{x}\) and \(\overline{y}\) are both 0, and \(\overline{z}<0\)

3.  If \(x = uv - 2u\) and \(y = uv\), compute the Jacobian.

a.  \(2u\)

b.  \(2uv\)

c.  \(2u^2v^2\)

d.  \(2u^2v\)

e.  \(2uv^2\)