# How do you find the area of the solid enclosed by the graphs #z=25(x^2+y^2)# and #z=8#?

##### 1 Answer

The answer is:

this function is a circular paraboloid, with the center in the origin of axes, and it is concave up.

The volume requested can be calculated with this triple integral:

where

The explanation is that this integral:

is a 4-dimension volume with "basis"

For this calculus we have to change the coordinate system. Instead of the cartesian coordinates, we have to use the cylindrical polar coordinates, that are:

and we have to remember that we have to calculate the Jacobian Matrix of the passing from a system of coordinates to an other, that is:

Since

our function becomes:

We have to find the volume under the plane

Since

but

The volume is in every part of every circle, sections of

Finally the integral becomes: