Triple Integrals over z-Simple Regions

Proposition (Triple Integrals Over z-Simple Regions) Suppose is a solid region bounded below by the surface and above by the surface that projects onto the region in the -plane. If is either type I (vertically simple) or type II (horizontally simple region), then the triple integral of the continuous function over is

triple integrals over z simple regions _gr_9.gif]

Proposition (Volume and Triple Integrals) A double integral can be interpreted as the area of the region of integration, via: if over a region in the and is the area of then

Similarly, we can interpret the triple integral as the volume of a solid; that is, if is the volume of the solid region then

Example (Volume as a Triple Integrals) Compute the volume of the tetrahedron bounded by the planes and

    Solution. The vertices of the tetrahedron are and So the region of integration in the -plane is bounded by the lines and This is determined by the vertices of the tetrahedron in the -plane and by determining the equations of the lines through these vertices.

triple integrals over z simple regions _gr_32.gif]

So the upper boundary is the plane that is Therefore,


Example (Volume as a Triple Integrals) Compute the volume of the solid region bounded above by the paraboloid and below by

    Solution. The bounded region as projection in the -plane given by the equation that is, and the graph of the regions are

triple integrals over z simple regions _gr_45.gif]

The volume is given by

Example (Volume as a Triple Integrals) Compute the volume of the solid bounded below by the paraboloid and above by the plane

    Solution. The projection of the bounded region onto the -plane is the graph of the equation by completing the square in the region is Therefore, the volume is given by

triple integrals over z simple regions _gr_58.gif]

which is not so easy to integrate. However, we can project this bounded region onto the -plane to obtain the region (in the -plane)

bounded by (take in ) and The volume of the bounded region is given by

Example (Volume as a Triple Integrals) Find the volume of the ellipsoid using triple integration.
triple integrals over z simple regions _gr_71.gif]
where
    Solution. We have and and so the volume is