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...Math multivariable calculus final exam solutions change the order of integration z zz y f x dxdydz dydzdx solution region is respectively integral equal to in what proportion surface sphere radius r divided by a plane passing at distance d from center justify your answer there are several ways compute area part above graph function pr over domain found integrating q s p zx zy polar coordinates arctanx we nd since and hence b alternatively spherical this piece given formula arccosd sin c perhaps most illuminating way based on representing as revolution about axis with cylindrical equation whose dz rest has thus required ratio table square rubber u v glued according rule find covered jacobian xu xv yu yv image times i e maximum minimum values xy unit circle lagrange multiplier method get system two linear equations which under constraint only when or have re spectively respective critical points corresponding therefore value does exist satisfying fxx fxy fyy local point neither nor each ...