Consider the equation below. 20x2 − y + 4z2 = 0 Reduce the equation to one of the standard forms   Sketch the surface (Write an equation for the cross section at z = 0 using x and y.)   (Write an equation for the cross section at y = −2 using x and z.)   (Write an equation for the cross section at y = 2 using x and z.)   (Write an equation for the cross section at x = 0 using y and z.)  

Consider the surface described by (x^2+y^2+z^2+A^2 - a^2)^2 = 4A^2(12 + y^2) where A > a(a) Using cylindrical coordinates, find a simple equation for the cross-section of the surface in the i-z-plane (that is. an arbitrary plane of constant theta) and sketch the cross-section. (b) Using the result of (a), find an equation in cylindrical coordinates for the cross-section (or cross sections) of the surface in the xy-plane and sketch the cross-section. (c) Using the result of (a) and (b) identify the surface. (d) Using a triple integral in cylindrical coordinates, find the volume inside the surface.

APPLICATION 1 (Engineering): DATA: The cross section of a tunnel is shown below in Fig.1 - The upper part of the cross section from A to B is given by the function f(x)9 B and A respectively. From E to C and F to D the cross section is vertical. 8 - From B to D and A to C the cross section follows the tangent lines to (x) at the points - A. Fig. 1 The tunnel cross section. All dimensions are in meters. FIND: 1. Find the vertical dimensions (i.e. the difference in height between points D & F, B & D, and G & B) of the cross section and write their values on fig.1 - Find the y coordinate of point G - Find the y coordinate of point B: Write the equation of the tangent line that passes through B 이