Sketch the solid whose volume is given by the iterated integ

Sketch the solid whose volume is given by the iterated integral below, and find the volume.

Solution

The solid described by the volume is the eighth of the sphere in the positive x, y, z octant. You can see this by noticing that the surface is sqrt(1-x^2-y^2), which is the top half of the sphere. Then the limits on y are from 0 to sqrt(1-x^2), which is the upper half of the unit circle. The limits on x are from 0 to 1, so just positive x-values. The volume of an eighth of a unit sphere is 1/8(4/3 pi) = pi/6. The instructions don\'t require you to use an integral to find the volume, so why bother! Just use your knowledge of spheres. :)