Understanding Non-Rectangular Regions for Integration

1. Suppose we want to integrate f(x,y) over a region D which is not necessarily a rectangle but can be contained inside a rectangle R. It will be helpful to classify certain types of non-rectangular regions. Consider a non-rectangular region D which can be described as where 1[a,b]R and 2[a,b]R are functions with 1(x)  2(x) . (x,y) such that a x  b and 1(x)  y  2(x) y y y x x x a b a b a b Such regions are called (by the text) y-simple. Some examples: (x,y) such that 1  x  3 and 2 – x  y  x2 (x,y) such that 1  x  3 and 1 – x  y  x2 (x,y) such that 4  x  9 and (6 + x)/5  y   x

2. Consider a non-rectangular region D which can be described as (x,y) such that c y  d and 1(y)  x  2(y) where 1[c,d]R and 2[c,d]R are functions with 1(y)  2(y) . y d Such regions are called (by the text) x-simple. c x Can the region (x,y) such that 4  x  9 and (6+x)/5  y   x be described easily as an x-simple region? Can the region (x,y) such that 1  x  3 and 1 – x  y  x2 be described easily as an x-simple region? Can the region (x,y) such that 1  x  3 and 2 – x  y  x2 be described easily as an x-simple region?

3. Note that the following regions are both x-simple and y-simple. Such regions are called (by the text) simple. y y y x x x Suppose we want to integrate the function f(x,y) over a y-simple region D, described by (x,y) such that a x  b and 1(x)  y  2(x), and suppose the rectangle R = [a,b][c,d] contains the region D. R (a,d) (b,d) y D (a,c) (b,c) x Defining the function f(x,y) to be equal to f(x,y) on the region D and equal to 0 (zero) outside the region D, we may write

4. b d f(x,y) dx dy = f(x,y) dx dy = DR f(x,y) dy dx = a c b 2(x) f(x,y) dy dx . a 1(x) (a,d) R (b,d) If the region D is x-simple and can be described by (x,y) such that c y  d and 1(y)  x  2(y), then we may write f(x,y) dy dx = D y d D 2(y) (a,c) (b,c) f(x,y) dx dy . x c 1(y) An x-simple and/or y-simple in region in R2 is called an elementary region.

5. Example Consider the following region D: y y = x2 (1/2 , 1/4) D x (1/2 , 0) Describe D as a y-simple region a x  b and 1(x)  y  2(x). Describe D as an x-simple region c y  d and 1(y)  x  2(y). 0  y  1/4 and  y  x  1/2 0  x  1/2 and 0  y  x2 Find the volume of the solid bounded by cylinder y = x2 and the planes z = 0, y = 0, x = 1/2, and x + y– z = 0. 3 / 160

6. Note that the area of any elementary region D in R2 can be found from dx dy = dy dx which can be written as a single integral of the difference between two functions of a single variable. D D y Example Consider the following region D: y = x2 Describe D as a y-simple region a x  b and 1(x)  y  2(x). (1/2 , 1/4) D 0  x  1/2 and x2  y  1/4 x (1/2 , 0) Describe D as an x-simple region c y  d and 1(y)  x  2(y). 0  y  1/4 and 0  x   y Find the area of the region D. 1 / 12

7. Example Consider the integral y xy dA D y = x3 where D is the following region: y = x2 (1 , 1) D Describe D as a y-simple region a x  b and 1(x)  y  2(x). x 0  x  1 and x3  y  x2 Describe D as a x-simple region c y  d and 1(y)  x  2(y). 0  y  1 and  y  x  3 y Find the double integral. 1/48

8. Example Find the area of the following region D: y The double integral which will give the area of D is y = 1/2 x2 (0 , 1) D dx dy (1/2 , 1/4) D y = x/2 The integration will be easiest when the region D is described as a region x y-simple a x  b and 1(x)  y  2(x). 1/2 1/2 x2 1/2 1/2 x2 1/2 1 dy dx = y dx = y = x/2 1/2 x2x/2 dx = x/2 0 0 0 1/2 x/2 x3 /3 x2 /4 = 7/48 x = 0