MATH 2374 Midterm: MATH 2374 UMN Spring 10 Exam 3sol
Document Summary
Third midterm: solutions: rewrite the given equation in the form x2. This equation de nes an ellip- soid centered at the origin with the x-, y- and z- radii equal toq 2. Such an ellipsoid can be parametrized by x( , ) =r2. 3 sin cos , y( , ) =r2. 3 sin sin , z( , ) = cos . (1) It is important to notice here that the parameter is not the angle between a radius-vector and the z-axis. The condition z x2 + y2 puts certain restrictions on . To determine them, we nd the intersection of the given ellipsoid and the surface1 z = x2 + y2: (3(x2 + y2) + 2z2 = 2 z = x2 + y2. We plug z = x2 + y2 into the rst equation and obtain 2z2 + 3z + 2 = 0. The positive solution of this quadratic equation is z = 1.
