Triple Integration

In this topic we work out many examples involving triple integration in a step-by-step manner. We state the necessary theorems and also work out triple integration examples in cylindrical and spherical coordinates. In most examples we sketch a graph of the region of integration in the -plane, explain how to visualize the solid region of integration, and how to setup the limits of integration.

Proposition (Triple Integration) Assume that all the given integrals exist on a rectangular region for given functions and

    (i)  Linearity Rule: For constants and
    

    
    (ii)   Dominance Rule:  If throughout a closed bounded region then
    


    (iii)   Subdivision Rule:  If the closed bounded region of integration is subdivided into two (disjoint) subdivisions and then

Proposition (Triple Integration) If is continuous over a rectangular box : then the triple integral may be evaluated by the iterated integral



The iterated integral can be performed in any order, with appropriate adjustments to the limits of integration.

Example (Triple Integration) Evaluate the triple integral where is the rectangular box

.

    Solution. We have
    









Example (Triple Integration) Evaluate the triple integral where is the rectangular box

.

    Solution. We have
    

Proposition (Triple Integration) 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

Example (Triple Integration) 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.



So the upper boundary is the plane that is Therefore,
    










Example (Triple Integration) 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



The volume is given by









Example (Triple Integration) 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
    


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 (Triple Integration) Find the volume of the ellipsoid using triple integration.

where
    Solution. We have and and so the volume is
    

Proposition (Triple Integration in Cylindrical Coordinates) Let be a region with upper surface and lower surface and let be the projection of the solid onto the -plane expressed in polar coordinates. Then, if is continuous on the triple integral of over is given by

Example (Triple Integration in Cylindrical Coordinates) Find the volume of the solid bounded by the paraboloid and the -plane.

    Solution. In the -plane we have the region Thus in cylindrical coordinates we have,
    

Example (Triple Integration in Cylindrical Coordinates) Use cylindrical coordinates to compute the integral



where is the solid bounded above by the plane and below by the surface

    Solution. We consider the region of integration as being -simple by projecting onto the -plane; and in the -plane we have the region bounded by

    

and so in cylindrical coordinates we have,
    

Example (Triple Integration in Cylindrical Coordinates) Use cylindrical coordinates to compute the integral



where is the cylindrical solid with

    Solution. We consider the region of integration as being -simple by projecting onto the -plane; and in the -plane we have the region   Noticing that



In cylindrical coordinates we have,
    

Example (Triple Integration in Cylindrical Coordinates) Use cylindrical coordinates to compute the integral



    Solution. The region of integration is





In cylindrical coordinates we have,
    

Proposition (Triple Integration in Spherical Coordinates) If is continuous on the closed bounded region , then the triple integral of over is given by





where is the region expressed in spherical coordinates.

Example (Triple Integration in Spherical Coordinates) Using spherical coordinates verify that the volume of a sphere is where is the radius of the sphere.

    Solution. Since the equation of the sphere is for and Then the volume is given by the triple integral in spherical coordinates,

Example (Triple Integration in Spherical Coordinates) Evaluate



where is the solid hemisphere and

    Solution. Using spherical coordinates we have,

Example (Triple Integration in Spherical Coordinates) Evaluate



where is the solid hemisphere

    Solution. Converting to spherical coordinates we have,

Example (Triple Integration in Spherical Coordinates) Evaluate



    Solution. Converting to spherical coordinates we have,

Example (Triple Integration in Spherical Coordinates) Find the volume of the solid region that remains in the spherical solid after the solid cone has been removed.

    Solution. We have

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