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Section 16.3 Triple Integrals • A continuous function of 3 variables can be integrated over a solid region, W, in 3-space just as a function of two variables can be integrated over a flat region in 2-space • We can create a Riemann sum for the region W – This involves breaking up the 3D space into small cubes – Then summing up the functions value in each of these cubes •If W {( x , y , z ) a x b , c y d , g z h } then n m f ( x , y , z ) dV lim f ( x i , y j , z k ) x y z m n k 1 j 1 i 1 p W p x ba n , y d c m ' z hg p •In this case we have a rectangular shaped box region that we are integrating over • We can compute this with an iterated integral – In this case we will have a triple integral f ( x , y , z ) dV h d g c b f ( x , y , z ) dx dy dz a W • Notice that we have 6 orders of integration possible for the above iterated integral • Let’s take a look at some examples Example • Find the triple integral f ( x, y, z ) e x y z W is the rectangular box with corners at (0,0,0), (a,0,0), (0,b,0), and (0,0,c) Example • Sketch the region of integration 1 0 1 z 1 1 2 f ( x , y , z ) dy dz dx 0 Example • Find limits for the integral W f ( x , y , z ) dV where W is the region shown z z y x y x This is a quarter sphere of radius 4 z z x y x y Triple Integrals can be used to calculate volume • Find the volume of the region bounded by z = x + y, z = 10, and the planes x = 0, y = 0 • Similar to how we can use double integrals to calculate the area of a region, we can use triple integrals to calculate volume – We will set f(x,y,z) = 1 Example • Find the volume of the pyramid with base in the plane z = -6 and sides formed by the three planes y = 0 and y – x = 4 and 2x + y + z =4. Example • Calculate the volume of the figure bound by the following curves x y 16 2 2 y 3 z 3 y z 3 2 y Some notes on triple integrals • Since triple integrals can be used to calculate volume, they can be used to calculate total mass (recall Mass = Volume * density) and center of mass • When setting up a triple integral, note that – The outside integral limits must be constants – The middle integral limits can involve only one variable – The inside integral limits can involve two variables