(937, #3) Find a parametric representation for the part of the elliptic paraboloid y = 6 - 3 x² - 2 z² that lies to the right of the xz-plane.

Solution: Note that for fixed y, the corresponding trace is a circle or an ellipse. So, write the equation as 3 x² + 2 z² = 6 - y or

for y = 0..6.This is an ellipse and has the parametric form

These are the equations used to create the graph with Maple:

plot3d([sqrt(2)*r*cos(t), 6*(1-r^2), sqrt(3)*r*sin(t)], r = 0..1, t = 0..2*Pi,
grid = [10,25], color = blue, tickmarks = [2,3,2], labels = [`x`, `y`, `z`],
labelfont = [HELVETICA, BOLD, 10]);