Triple Integrals. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Triple Integrals

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1 Triple Integrals MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 211

2 Riemann Sum Approach Suppose we wish to integrate w f (x, y, z), a continuous function, on the box-shaped region {(x, y, z) a x b, c y d, m z n}. 1 Let P { k } N k1 be a partition of into rectangular boxes. 2 Let the dimensions of k be x k, y k, and z k, then V k x k y k z k. 3 Let (u k, v k, w k ) be any point in k.

3 Riemann Sum Approach Riemann Sum: N f (u k, v k, w k ) V k. k1 If P is the length of the longest box diagonal in P, then we may define the triple integral.

4 Riemann Sum Approach Riemann Sum: N f (u k, v k, w k ) V k. k1 If P is the length of the longest box diagonal in P, then we may define the triple integral. Definition For any function f (x, y, z) defined on the rectangular box, we define the triple integral of f over by f (x, y, z) dv lim P k1 N f (u k, v k, w k ) V k, provided the limit exists and is the same for every choice of evaluation points (u k, v k, w k ) in k.

5 Fubini s Theorem Theorem (Fubini s Theorem) Suppose that f (x, y, z) is continuous on the box defined by {(x, y, z) a x b, c y d, m z n}. Then we can write the triple integral over as the triple iterated integral: f (x, y, z) dv b d n a c m f (x, y, z) dz dy dx.

6 Fubini s Theorem Theorem (Fubini s Theorem) Suppose that f (x, y, z) is continuous on the box defined by {(x, y, z) a x b, c y d, m z n}. Then we can write the triple integral over as the triple iterated integral: f (x, y, z) dv b d n a c m f (x, y, z) dz dy dx. There are five other equivalent orders of integration.

7 Example (1 of 4) Let {(x, y, z) x 1, 1 y 2, z 3} and evaluate xyz 2 dv.

8 Example (2 of 4) xyz 2 dv xyz 2 dx dz dy 1 2 x 2 yz 2 1 dz dy 1 2 yz2 dz dy 9 2 y dy 9 4 y yz3 3 dy 27 4

9 Example (3 of 4) Let {(x, y, z) x 2, 3 y, 1 z 1} and evaluate (x 2 + yz) dv.

10 Example (4 of 4) (x 2 + yz) dv (x 2 + yz) dx dy dz ( ) x 3 + xyz dy dz ( ) yz dy dz ) dz ( 8 3 y + y 2 z 3 8 9z dz 8z 9 2 z2 1 1

11 Triple Integrals Over General Regions (1 of 2) To develop of the triple integral of f (x, y, z) over a general region we must form an inner partition of. y z x

12 Triple Integrals Over General Regions (2 of 2) Definition For a function f (x, y, z) defined in the bounded, solid region, the triple integral of f (x, y, z) over is f (x, y, z) dv lim P k1 N f (u k, v k, w k ) V k, provided the limit exists and is the same for every choice of evaluation points (u k, v k, w k ) in k.

13 Evaluating Triple Integrals If region can be described as {(x, y, z) (x, y) R, k 1 (x, y) z k 2 (x, y)} then f (x, y, z) dv R k2 (x,y) k 1 (x,y) f (x, y, z) dz da.

14 Example (1 of 7) Integrate f (x, y, z) z over the region bounded by the plane x + y + z 1 and the coordinate planes.

15 Example (2 of 7) 1. y z x 1.

16 Example (3 of 7) z dv R 1 x y R 1 1 x z dz da 1 2 (1 x y)2 da 1 1 (1 x y) 2 dy dx (1 x y) 3 1 x dx (1 x) 3 dx 1 24 (1 1 x)4

17 Example (4 of 7) Integrate f (x, y, z) x 2 + z 2 over the portion of the paraboloid y x 2 + z 2 where y 4. 4 y z x 1 2

18 Example (5 of 7) x 2 + z 2 dv 2π 2π R 4 R 2π 2 2 x 2 +z 2 x 2 + z 2 dy da 4 x 2 + z 2 (x 2 + z 2 ) 3/2 da 128π 15 (4r r 3 )r dr dθ (4r 2 r 4 ) dr ( 4 3 r 3 1 ) 2 5 r 5

19 Example (6 of 7) Find the volume of the solid region in the positive orthant bounded by z 2 y and x 4 y z y x 2.4

20 Example (7 of 7) V R dv (2 y) da 2 y R 2 4 y 2 (2 y)(4 y 2 ) dy 1 dz da 2 ( 1 4 y 4 2 ) 2 3 y 3 2y 2 + 8y (2 y) dx dy (y 3 2y 2 4y + 8) dy

21 Mass and Center of Mass If ρ(x, y, z) denotes the density of material at (x, y, z) in region, then the mass of the solid occupying region is m ρ(x, y, z) dv. The moments of with respect to the coordinate planes are M yz xρ(x, y, z) dv M xz yρ(x, y, z) dv M xy zρ(x, y, z) dv The center of mass is the point with coordinates: ( Myz (x, y, z) m, M xz m, M ) xy m

22 Example (1 of 4) Find the mass and center of mass of the solid region bounded by the plane z 4 and the paraboloid z x 2 + y 2 whose density is described by ρ(x, y, z) 3 + x.

23 Example (2 of 4) 4 3 z x J. Robert Buchanan Triple Integrals -2 2 y

24 Example (3 of 4) m R 2π 2 2π 2 2π 24π (3 + x) dv R 4 [ 4(3 + x) (x 2 + y 2 )(3 + x) [ 12 + (3 + x) dz da x 2 +y 2 ] da [ ] 4(3 + r cos θ) r 2 (3 + r cos θ) r dr dθ [12r 3r 3 + (4r 2 r 4 ) cos θ] dr dθ ( ) ] cos θ dθ 5

25 Example (4 of 4) M yz M xz M xy (3x + x 2 ) dv 16π 3 (3y + xy) dv (3z + xz) dv 64π Thus (x, y, z) ( Myz m, M xz m, M ) ( xy 2 m 9,, 8 ). 3

26 Homework Read Section Exercises: 1 43 odd