15.3: Double Integrals over General Regions

Transcription

1 15.3: ouble Integrals over General Regions Marius Ionescu November 19, 2012 Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

2 ouble Integrals over General Regions Fact We want to integrate a function f over bounded regions of more general shape: Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

3 First things rst: enition enition If is a bounded region, then we dene a new function F with domain a rectangle R that contains by { f (x, y) if (x, y) is in F (x, y) = 0 otherwise Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

4 enition Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

5 enition enitions The double integral of f over is f (x, y)da = F (x, y)da. R Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

6 omains of type I enition A domain is of type I if it lies between the graphs of two continuous functions of x: = {(x, y) a x b, g 1 (x) y g 2 (x)} Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

7 ouble Integrals over omains of type I Fact If is a region of type I and f is continuous then f (x, y)da = a ˆ b ˆ g2(x) g1(x) f (x, y)dydx. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

8 Evaluate the following double integrals: Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

9 Evaluate the following double integrals: y x 5 da, where = {(x, y) 0 1, 0 y x 2 } +1 Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

10 Evaluate the following double integrals: y x 5 da, where = {(x, y) 0 1, 0 y x 2 } +1 (x + 2y)dA, where is the region bounded by the parabolas y = 2x 2 and y = 1 + x 2. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

11 Evaluate the following double integrals: y x 5 da, where = {(x, y) 0 1, 0 y x 2 } +1 (x + 2y)dA, where is the region bounded by the parabolas y = 2x 2 and y = 1 + x 2. (x 2 + 2y)dA, where is bounded by y = x, y = x 2,x 0. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

12 omains of type II enition A domain is of type II if it can be expressed as = {(x, y) : c y d, h 1 (y) x h 2 (y)}. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

13 ouble Integrals over omains of type II Fact If is a region of type II and f is continuous then f (x, y)da = c ˆ d ˆ h2(y) h1(y) f (x, y)dxdy. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

14 Evaluate the following integrals Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

15 Evaluate the following integrals xyda, where is the region bounded by the line y = x 1 and the parabola y 2 = 2x + 6. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

16 Evaluate the following integrals xyda, where is the region bounded by the line y = x 1 and the parabola y 2 = 2x + 6. y 2 e xy da, where is the region bounded by y = x, y = 4, x = 0. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

17 More Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

18 More Evaluate the iterated integral x sin(y 2 )dydx. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

19 More Evaluate the iterated integral x sin(y 2 )dydx. Find the volume of the tetrahedron bounded by the planes x + 2y + z = 2, x = 2y, x = 0, and z = 0. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

20 More Evaluate the iterated integral x sin(y 2 )dydx. Find the volume of the tetrahedron bounded by the planes x + 2y + z = 2, x = 2y, x = 0, and z = 0. Find the volume of the solid under the surface z = 1 + x 2 y 2 above the region enclosed by x = y 2 and x = 4. and Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

21 Properties of the double integral Fact Properties of the double integral: [f (x, y) + g(x, y)]da = f (x, y)da + cf (x, y)da = c f (x, y)da. If f (x, y) g(x, y) for all (x, y) in, then f (x, y)da g(x, y)da. g(x, y)da. If = 1 2, where 1 and 2 don't overlap except perhaps on their boundaries, then f (x, y)da = f (x, y)da + f (x, y)da. 1 2 Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

22 Properties of the double integral (cont'd) Fact Properties of the double integral: 1dA = area of = A(). If m f (x, y) M, then ma() f (x, y)da MA(). Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

23 Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

24 Find the are of the triangle with vertices (0, 0), (5, 0), and (5, 4) (using double integrals). Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15

25 Find the are of the triangle with vertices (0, 0), (5, 0), and (5, 4) (using double integrals). Estimate the integral e sin x cos y da, where is the disk with center the origin and radius 2. Marius Ionescu () 15.3: ouble Integrals over General Regions November 19, / 15