Section 16.7Triple Integrals

TRIPLE INTEGRAL OVER A BOXConsider a function w = f (x, y, z) of three variables defined on the box B given by

Divide B into sub-boxes by dividing the interval [a, b] into l subintervals of equal width x, dividing [c, d] into m subintervals of equal width y, and dividing [r, s] into n subintervals of equal width z. This divides the box B into lmn sub-boxes. A typical sub-box Bilk isBijk = [xi 1, xi] [yj 1, yj] [zk 1, zk]Each sub-box has volume V = x y z.

TRIPLE INTEGRAL OVER A BOX (CONTINUED)We now form the triple Riemann sum

where the sample point is in box Bijk.

TRIPLE INTEGRAL OVER A BOX (CONCLUDED)The triple integral of f over the box B is

if this limit exists.The triple integral always exists if f is continuous. If we choose the sample point to be (xi, yj, zk), the triple integral simplifies to

FUBINIS THEOREM FOR TRIPLE INTEGRALTheorem: If f is continuous on the rectangular box B = [a, b] [c, d] [r, s], thenNOTE: The order of the partial antiderivatives does not matter as long as the endpoints correspond to the proper variable.

EXAMPLEEvaluate the triple integral , whereB is the rectangular box given byB = {(x, y, z) | 1 x 2, 0 y 1, 0 z 2}

TRIPLE INTEGRAL OVER A BOUNDED REGIONThe triple integral over the bounded region E is defined as

where B is a box containing the region E and the function F is defined as

TYPE 1 REGIONSThe region E is said to by of type 1 if it lies between to continuous functions of x and y. That is,

where D is the projection of E onto the xy-plane.The triple integral over a type 1 region is

TYPE 1 REGIONS (CONTINUED)If D is a type I region in the xy-plane, then E can be described as

and the triple integral becomes

TYPE 1 REGIONS (CONCLUDED)If D is a type II region in the xy-plane, then E can be described as

and the triple integral becomes

EXAMPLEEvaluate the triple integral , whereE is the region bounded by the planes x = 0, y=0, z = 0, and 2x + 2y + z = 4.

TYPE 2 REGIONSThe region E is said to by of type 2 if it lies between two continuous functions of y and z. That is,

where D is the projection of E onto the yz-plane.The triple integral over a type 2 region is

TYPE 3 REGIONSThe region E is said to by of type 3 if it lies between two continuous functions of x and z. That is,

where D is the projection of E onto the xz-plane.The triple integral over a type 3 region is

EXAMPLEEvaluate the triple integral , where E is the region bounded by the paraboloid x = y2 + z2 and the plane x = 4.

VOLUME AND TRIPLE INTEGRALSThe triple integral of the function f (x, y, z) = 1 over the region E gives the volume of E; that is,

EXAMPLEFind the volume of the region E bounded by the plane z = 0, the plane z = x, and the cylinder x=4 y2.

MASSSuppose the density function of a solid object that occupies the region E is (x, y, z). Then the mass of the solid is

MOMENTSSuppose the density function of a solid object that occupies the region E is (x, y, z). Then the moments of the solid about the three coordinate planes are

CENTER OF MASSThe center of mass is located at the point whereIf the density is constant, the center of mass of the solid is called the centroid of E.

MOMENTS OF INERTIAThe moments of inertia about the three coordinate axis are

EXAMPLEFind the mass and center of mass of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 1 whose density function is given by (x, y, z) = y.