Graph the solid bounded by the parabolic cylinder z 4 y2 a

Graph the solid bounded by the parabolic cylinder z = 4 - y^2 and the planes x + 2z = 1, x = 0 and z = 0. write E as a type I, II or III region (E = {find what goes here}).

Solution

I assume that E is the solid from Part a).

Observe that E looks like a wedge placed on the xy-plane, and its top surface is a plane (x+2z=1) bounded in the z-direction by the parabolic cylinder. So one can evaluate the volume of E as the integral of the top surface plane from x=0 to the plane, z=0 to the plane, and y between the positive and negative square roots of z. In this case, z is a function of x and y, and so E is a type I region.