Double integrals over a General (Bounded) Region

Transcription

1 ouble integrals over a General (Bounded) Region Recall: Suppose z f (x, y) iscontinuousontherectangularregionr [a, b] [c, d]. We define the double integral of f on R by R f (x, y) da lim max x i, y j!0 mx i1 nx j1 f (x ij, y ij ) A ij. Question: What if the domain of f or the region on which we integrate f is not rectangular? Suppose is a bounded region including all the points on its boundary Boundedness means is a subset of some rectangular region R [a, b] [c, d] Figure: Placing inside some R Math 67 (University of Calgary) Fall 015, Winter / 13

2 ouble integrals over a General (Bounded) Region f f is defined and is continuous on, wedefineanewfunction f on R to agree with f on and equal 0elsewhereonR. f (x, y) f (x, y), (x, y) 0, (x, y) / Figure: Place inside some R; Extendf to f Although f could be discontinuous along the boundary of, onecanprovethatthedefinitionof f (x, y) da remains valid provided the boundary of is su ciently nice R Now, we define f (x, y) da f (x, y) da R Math 67 (University of Calgary) Fall 015, Winter 016 / 13

3 terated integrals To compute RR f (x, y) da for a general bounded region, Fubini smethodofiteratedintegrationworksto produce the slicing formula: Suppose that is bounded above by y h(x) and below by y g(x) and x a and x b. Then Z xb Z! yh(x) f (x, y) da f (x, y) dy dx. xa yg(x) Figure: Finding double integrals by computing iterated integrals Suppose that is bounded to the right by x h(y) and to the left by x g(y) and y c and y d. Then Z yd Z! xh(y) f (x, y) da f (x, y) dx dy. yc xg(y) Math 67 (University of Calgary) Fall 015, Winter / 13

4 Properties of double integrals Some properties of double integrals (assuming all integrals exist): (f (x, y) +g(x, y)) da cf (x, y) da c f (x, y) da + f (x, y) da g(x, y) da f we split up into two regions 1 [ with no overlap (except at possibly the boundary), then f (x, y) da f (x, y) da + f (x, y) da 1 Figure: Splitting the domain of integration Compare with splitting [a, c] into[a, b] [ [b, c] in R c a f (x) dx R b a f (x) dx + R c f (x) dx b Math 67 (University of Calgary) Fall 015, Winter / 13

5 Special cases: Area and Volume, Length and Area: n Calculus, the net area under f (x) over[a, b] is f (x) dx a n Calculus, the net volume under f (x, y) over is f (x, y) da n Calculus, the length of [a, b] is n Calculus, the area of is Guidelines for evaluating double integrals: Sketch the region Slice - Z b a 1 da horizontally vertically radially (later, using polar coordinates) Set up the iterated integral and evaluate 1 dx Z b Math 67 (University of Calgary) Fall 015, Winter / 13

6 Example Evaluate yx da where is the bounded region enclosed by y x and y p x. Solution: We first find the intersection points of the two graphs. x p x ) x 4 x ) x 4 x 0 ) x(x 3 1) 0 ) x 0orx 3 1 ) x 0orx 1. Figure: The intersection of y x and y p x Slice vertically at x (0 apple x apple 1), the slice enters the region at the bottom curve y x and leaves the region at the top curve y p x. The region is given by x apple y apple p x, 0 apple x apple 1. Math 67 (University of Calgary) Fall 015, Winter / 13

7 Figure: Using vertical slicing, order of integration RR (...) dy dx f (x, y) da 0 Z y p x yx p x xy dy! dx x 1 C x A dx x y! y p x yx x dx x 5! dx x 3 6! x 6 x Math 67 (University of Calgary) Fall 015, Winter / 13

8 Using the order dxdy The graphs intersect at x 0andx 1,andthecorrespondingy-coordinates are y 0andy 1 Slicing horizontally at y (0 apple y apple 1), the slice enters from the left curve y p x (or, x y )and leaves at the right curve y x (or, since x 0, x p y) Figure: Using horizontal slicing, order of integration RR (...) dx dy is described by y apple x apple p y,0apple y apple 1. f (x, y) da Z y1 Z p x y y0 xy Z y1 xy dx dy y0 x y! x p y xy dy Math 67 (University of Calgary) Fall 015, Winter / 13

9 Order of integration Sometimes we want to switch the order of integration (dxdy to dydx or dydx to dxdy). Example Evaluate Z 1 Z 1. Solution: 0 y e x dx dy We cannot integrate e x (in terms of a finite number of elementary functions) with respect to x. We switch order. From the given iterated integral, the region is y apple x apple 1, 0 apple y apple 1. Figure: Change of direction of slicing; Change of order of integration Math 67 (University of Calgary) Fall 015, Winter / 13

10 Figure: Change of direction of slicing; Change of order of integration The current order (dxdy) isbasedonhorizontalslicesofthetriangularregion. The horizontal slices enter from the left curve x y and leave at the right curve x 1. Let s slice vertically. The vertical slices enter from the bottom curve y 0andleaveatthetopcurvey x. has description 0 apple y apple x, 0 apple x apple 1. Z y1 e x dx dy y0 xy Z yx y0 e x dy dx Math 67 (University of Calgary) Fall 015, Winter / 13

11 Z y1 y0 xy e x dx dy (use substitution u x ) Z yx y0 e x dy dx yx e x y dx y0 e x x e 0 0 dx e x x dx 1 x1 ex 1 1 e1 e0 e 1 Math 67 (University of Calgary) Fall 015, Winter / 13

12 Example Compute R (x y) da, where R is the bounded region enclosed by x 0, x 1, y x and y x. Solution: Sketch R. Figure: Region of integration: x apple y apple x, 0 apple x apple 1 Slice vertically at x (0 apple x apple 1). The slices enter R from y x and leave R at y x. (x y) da R Z y x yx 00 x (x y) dy! dx x 1 C A xy x (x)! y y x x yx dx! 1 C A dx Math 67 (University of Calgary) Fall 015, Winter / 13

13 Further examples/exercises 1 Find the values of the following double integrals. e x da, where is the bounded region enclosed by the curves y 1 x, y 1+x, x,andx 3. 1 y da, where is the bounded region enclosed by the curves y x, x 1, +1 and y 0. fpossible,trybothordersofintegration. y sin (x ) da, where (x, y) :y apple x apple p, 0 apple y apple 1/4.fpossible,try both orders of integration. yda,where is the region bounded by the unit circle x + y 1. Find the volume of the solid under the surface z x + y and above the region on the xy-plane bounded by y x and x y 3 in the first quadrant. Z 1 Z 1 3 Find the iterated integral p dy dx. x/3 y Math 67 (University of Calgary) Fall 015, Winter / 13