Question

Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 6.

Answer 1

With x = 0, y = 0, and z = 0 we find ourselves in the corner of a cube. x + y + z = 6 gives us a sphereical shape with a radius of 6 centered at the origin (0,0,0).

On an x-y graph, the axes break the region into four quadrants. On an x-y-z graph, we have 8 quadrants (octorants?). Because we are bounded by the planes x=y=z=0, we are only concerned with the one quadrant where all are positive. Therefore we are looking at the volume of 1/8 of a sphere with radius 6

Area = (4/3)pi*r^3 = (4/3)pi*216 = 288 pi = 904.32