Suppose we want to integrate fx y over

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 (x, y) such that a x b and 1(x) y 2(x) where 1[a, b] R and 2[a, b] R are functions with 1(x) 2(x). y y y x x a b Such regions are called (by the text) y-simple. a b Some examples: (x, y) such that 1 x 3 and 2 – x y x 2 (x, y) such that 1 x 3 and 1 – x y x 2 (x, y) such that 4 x 9 and (6 + x)/5 y x x

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 x 2 be described easily as an x-simple region? Can the region (x, y) such that 1 x 3 and 2 – x y x 2 be described easily as an x-simple region?

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) (a, d) R y over a y-simple region D, described by (x, y) such that a x b and 1(x) y 2(x), D and suppose the rectangle R = [a, b] [c, d] (a, c) contains the region D. (b, d) (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

b f(x, y) dx dy D b = d f(x, y) dx dy = R f(x, y) dy dx = a 2(x) c f(x, y) dy dx. a 1(x) 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 d 2(y) f(x, y) dy dx = f(x, y) dx dy. D c 1(y) An x-simple and/or y-simple in region in R 2 is called an elementary region. (a, d) R (b, d) y D (a, c) (b, c) x

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

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

Example Consider the integral xy d. A D where D is the following region: y y = x 3 y = x 2 (1 , 1) Describe D as a y-simple region a x b and 1(x) y 2(x). 0 x 1 and x 3 y x 2 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 D x

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