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Section 16.7 Triple Integrals
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Section 16.7. Triple Integrals. TRIPLE INTEGRAL OVER A BOX. Consider a function w = f ( x , y , z ) of three variables defined on the box B given by - PowerPoint PPT Presentation

### Transcript of Section 16.7

• 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.