Riemann sum and double integral.

� 22. Double integrals Definition 22.1. Let R =[a,b] × [c,d] ⊂ R2 be a rectangle in the plane. A partition P of R is a pair 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. We write ��f(x,y)dx dy = I,R to mean that f is integrable with integral I. We use a sneaky trick to integrate over regions other than rectangles. Suppose that D is a bounded subset of the plane. Then we can find a rectangle R which completely contains D. Definition 22.4. The indicator function of D ⊂ R is the function iD : R 1−→ R, � �� � �� �� given by iD(x)= 1 if x ∈ D 0 if x/∈ D. If iD is integrable, then we say that the area of D is the integral iD dx dy. R If iD is not integrable, then D does not have an area. Example 22.5. Let D = { (x,y) ∈ [0, 1] × [0, 1] | x,y ∈ Q }. Then D does not have an area. Definition 22.6. If f : D −→ R is a function and D is bounded, then pick D ⊂ R ⊂ R2 a rectangle. Define f˜: R −→ R, by the rule f˜(x)= f(x) if x ∈ D 0 otherwise. We say that f is integrable over D if f˜ is integrable over R. In this case �� �� f(x,y)dx dy = f˜(x,y)dx dy. D R Proposition 22.7. Let D ⊂ R2 be a bounded subset and let f : D −→ R and g : D −→ R be two integrable functions. Let λ be a scalar. Then (1) f + g is integrable over D and �� �� �� f(x,y)+ g(x,y)dx dy = f(x,y)dx dy + g(x,y)dx dy. D DD (2) λf is integrable over D and λf(x,y)dx dy = λf(x,y)dx dy. D D (3) If f(x,y) ≤ g(x,y) for any (x,y) ∈ D, then �� �� f(x,y)dx dy ≤ g(x,y)dx dy. D D 2 �� �� (4) |f| is integrable over D and | D f(x,y)dx dy|≤ D|f(x,y)| dx dy. It is straightforward to integrate continuous functions over regions of three special types: Definition 22.8. A bounded subset D ⊂ R2 is an elementary region if it is one of three types: Type 1: D = { (x,y) ∈ R2 | a ≤ x ≤ b,γ(x) ≤ y ≤ δ(x) }, where γ :[a,b] −→ R and δ :[a,b] −→ R are continuous functions. Type 2: D = { (x,y) ∈ R2 | c ≤ y ≤ d,α(y) ≤ x ≤ β(y) }, where α :[c,d] −→ R and β :[c,d] −→ R are continuous functions. Type 3: D is both type 1 and 2. Theorem 22.9. Let D ⊂ R2 be an elementary region and let f : D −→ R be a continuous function. Then (1) If D is of type 1, then �� � b �� δ(x) � f(x,y)dx dy = f(x,y)dy dx. D aγ(x) (2) If D if of type 2, then �� � d �� β(y) � f(x,y)dx dy = f(x,y)dx dy. D cα(y) Example 22.10. Let D be the region bounded by the lines x =0, y =4 and the parabola y = x2 . Let f : D −→ R be the function given by f(x,y)= x2 + y2 . 3 If we view D as a region of type 1, then we get �� � 2 �� 4 � f(x,y)dx dy = x 2 + y 2 dy dx D 0 x2 � 2� 3 �4 = x 2 y + ydx 30 x2 � 2 26 x6 =4x 2 +3 − x 4 − 3dx 0� 4x3 26xx5 x7 �2 = 3+3 − 5 − 37· 0 25 27 25 27 =3+3 − 5 − 37· 26 28 =+ 35 7·�� 122 =26 + . 35 7· On the other hand, if we view D as a region of type 2, then we get �� � 4 �� √y � f(x,y)dx dy = x 2 + y 2 dx dy D 00 � 4� 3 �√yx=+ xy 2 dy300� 4 y3/2 =+ y 5/2 dy30 ��4 2y5/2 2y7/2 =+ 35 7· 0 26 28 =+ 35 7·�� 122 =26 + . 35 7· 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.