� � � 24. Triple integrals Definition 24.1. Let B =[a,b] × [c,d] × [e,f] ⊂ R3 be a box in space. A partition P of R is a triple of sequences: a = x0 0, we may find a δ> 0 such that for every mesh P whose mesh size is less than δ, we have |I − S| < �, where S is any Riemann sum associated to P. If W ⊂ R3 is a bounded subset and f : W −→ R is a bounded function, then pick a box B containing W and extend f by zero to a function f˜: B −→ R, f˜(x)= x if x ∈ W 0 otherwise. If f˜ is integrable, then we write��� ��� f(x,y,z)dx dy dz = f˜(x,y,z)dx dy dz. W B In particular ��� vol(W )= dx dy dz. W There are two pairs of results, which are much the same as the results for double integrals: 1 | | ��� ��� ��� Proposition 24.4. Let W ⊂ R2 be a bounded subset and let f : W −→ R and g : W −→ R be two integrable functions. Let λ be a scalar. Then (1) f + g is integrable over W and ��� ��� ��� f(x,y,z)+g(x,y,z)dx dy dz = f(x,y,z)dx dy dz+ g(x,y,z)dx dy dz. W WW (2) λf is integrable over W and ��� ��� λf(x,y,z)dx dy dz = λf(x,y,z)dx dy dz. WW (3) If f(x,y,z) ≤ g(x,y,z) for any (x,y,z) ∈ W , then ��� ��� f(x,y,z)dx dy dz ≤ g(x,y,z)dx dy dz. WW (4) fis integrable over W and | f(x,y,z)dx dy dz|≤ |f(x,y,z)| dx dy dz. WW Proposition 24.5. Let W = W1 ∪ W2 ⊂ R3 be a bounded subset and let f : W −→ R be a bounded function. If f is integrable over W1 and over W2, then f is integrable over W and and W1 ∩ W2, and we have ��� ��� ��� f(x,y,z)dx dy dz = f(x,y,z)dx dy dz+ f(x,y,z)dx dy dz WW1 W2 − f(x,y,z)dx dy dz. W1∩W2 Definition 24.6. Define three maps πij : R3 −→ R2 , by projection onto the ith and jth coordinate. In coordinates, we have π12(x,y,z)=(x,y),π23(x,y,z)=(y,z), and π13(x,y,z)=(x,z). For example, if we start with a solid pyramid and project onto the xy-plane, the image is a square, but it project onto the xz-plane, the image is a triangle. Similarly onto the yz-plane. Definition 24.7. A bounded subset W ⊂ R3 is an elementary subsse if it is one of four types: Type 1: D = π12(W ) is an elementary region and W = { (x,y,z) ∈ R2 | (x,y) 2∈ D,�(x,y) ≤ z ≤ φ(x,y) }, where �: D −→ R and φ: D −→ R are continuous functions. Type 2: D = π23(W ) is an elementary region and W = { (x,y,z) ∈ R2 | (y,z) ∈ D,α(y,z) ≤ x ≤ β(y,z) }, where α : D −→ R and β : D −→ R are continuous functions. Type 3: D = π13(W ) is an elementary region and W = { (x,y,z) ∈ R2 | (x,z) ∈ D,γ(x,z) ≤ y ≤ δ(x,z) }, where γ : D −→ R and δ : D −→ R are continuous functions. Type 4: W is of type 1, 2 and 3. The solid pyramid is of type 4. Theorem 24.8. Let W ⊂ R3 be an elementary region and let f : W −→ R be a continuous function. Then (1) If W is of type 1, then ��� �� �� φ(x,y) � f(x,y,z)dx dy dz = f(x,y,z)dz dx dy. Wπ12(W ) �(x,y) (2) If W if of type 2, then ��� �� �� β(y,z) � f(x, y, z) dx dy dz = f(x, y, z) dx dy dz. W π23(W ) α(y,z) (3) If W if of type 3, then ��� �� �� δ(x,z) � f(x,y,z)dx dy dz = f(x,y,z)dy dx dz. Wπ13(W ) γ(x,z) Let’s figure out the volume of the solid ellipsoid: W = { (x,y,z) ∈ R3 | � xa �2 + � yb �2 + � zc �2 ≤ 1 }. 3 � This is an elementary region of type 4. ��� vol(W )= dx dy dz W � a ⎛� bq1−(x )⎛� cq1−(xa )−(yb )⎞⎞ 2 22 a = ⎝ q⎝ qdz⎠ dy⎠ dx 2 22 −a −b1−(ax )−c1−(xa )−(yb )2� a ⎛� bq1−(x )�� x �2 � y �2 ⎞ a = ⎝ q2c dy⎠ dx 2 1 − a − b−a −b1−(xa )2� a ⎛� bq1−(x )� x �2 � y �2 ⎞ a =2c ⎝ q1 − dy⎠ dx 2 a − b−a −b1−(xa )� ⎛� q�� �⎞ 2c ab1−(xa )2 � x �2 = ⎝ qb22 dy⎠ dx1 −− yb −a −b1−(x )2 a a πc � a �� x �2 � = b2 1 − dx ba �−aa � x �2 = πbc dx1 − a �−a �3 a x= πbc x − 3a2 � −a�3a= πbc 2a − 2 3a24π= abc. 3 4 MIT OpenCourseWarehttp://ocw.mit.edu 18.022 Calculus of Several Variables Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.