Question

Find the average value of F(x, y, z) = z over the region bounded below by the xy-plane, on the sides by the sphere x² + y² + z² = 36, and bounded above by the cone ϕ = (pi/3).

Answer 1

The average value of z over the region is 3.

To find the average value of z over the given region, we can use geometric reasoning without performing the integral explicitly.

The region is bounded below by the xy-plane and bounded above by the cone
`\(\phi = (\pi)/(3)\)`, with the sphere
`\(x^2 + y^2 + z^2 = 36\)` in between.

For the cone
`\(\phi = (\pi)/(3)\)`, the upper limit of z is 6, as the maximum z value for the cone is
`\(r \cdot \tan(\phi) = 6\)`, where r = 6 and
`\(\phi = (\pi)/(3)\)`.

Since the lower limit of z is 0 and the upper limit is 6 within the region, the average value of z across this region is halfway between 0 and 6, which is
`\((6)/(2) = 3\)`.