Math 240: Double Integrals and Green s Theorem

Transcription

1 Math 240: Double Integrals and Green s Theorem yan Blair University of Pennsylvania Thursday March 17, 2011 yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

2 Outline 1 eview 2 Today s Goals 3 Double Integrals 4 Green s Theorem yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

3 eview eview of Last Time 1 Learned how to evaluate line integrals. 2 Learned how to test for path independence. yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

4 eview Theorem (Fundamental theorem of Line integrals) Suppose there exists a function φ(x, y) such that dφ = P(x, y)dx + Q(x, y)dy. Then Pdx + Qdy = φ(b) φ(a). yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

5 eview Test for path independence in 2D Theorem Let P and Q have continuous first partial derivatives in an open simply connected region. Then Pdx + Qdy is independent of path C C if and only if for all (x, y) in the region. P y = Q x yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

6 eview An Example Question Evaluate the following integral and verify it is path independent. (2,2) (0,0) (y 3 + 3x 2 y)dx + (x 3 + 3y 2 x + 1)dy yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

7 Today s Goals Today s Goals 1 eview how to evaluate double integrals in standard coordinates and polar coordinates. 2 Learn Green s Theorem and how to use it. yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

8 Double Integrals Intuition of line integrals in the plane Suppose we want to find the volume of an object with a flat base in the shape of the region in the plane. yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

9 Double Integrals Intuition of line integrals in the plane Suppose we want to find the volume of an object with a flat base in the shape of the region in the plane. Additionally, the sides of the object are vertical and the top of the object is the graph of the function G(x, y). yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

10 Double Integrals Intuition of line integrals in the plane Suppose we want to find the volume of an object with a flat base in the shape of the region in the plane. Additionally, the sides of the object are vertical and the top of the object is the graph of the function G(x, y). Then the volume of the object is given by G(x, y)da Where we are integrating with respect to the area of. yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

11 egions Double Integrals Definition A Type I region is given by the following formula a x b, g 1 (x) y g 2 (x) Definition A Type II region is given by the following formula c y d, h 1 (x) y h 2 (x) yan Blair (U Penn) Math 240: Double Integrals and Green s Theorem Thursday March 17, / 15

12 Double Integrals Evaluation of Double Integrals Theorem Let f be continuous on a region. If is Type I, then If is Type II, then f (x, y)da = f (x, y)da = b g2 (x) a g 1 (x) d h2 (y) c h 1 (y) f (x, y)dydx f (x, y)dxdy yan Blair (U Penn) Math 240: Double Integrals and Green s TheoremThursday March 17, / 15

13 Double Integrals Example For the region bounded by y = x, x + y = 4 and x = 0 evaluate x + 1dA yan Blair (U Penn) Math 240: Double Integrals and Green s TheoremThursday March 17, / 15

14 Double Integrals Example For the region given by 0 x 2, x 2 y 4 evaluate xe y2 da yan Blair (U Penn) Math 240: Double Integrals and Green s TheoremThursday March 17, / 15

15 Double Integrals Evaluation of Double Integrals in Polar Coordinates Theorem Let f be continuous on a region. If is Type PI, then If is Type PII, then f (r, θ)da = f (r, θ)da = β g2 (θ) α g 1 (θ) b h2 (r) a h 1 (r) f (r, θ)rdrdθ f (r, θ)rdθdr yan Blair (U Penn) Math 240: Double Integrals and Green s TheoremThursday March 17, / 15

16 Double Integrals Change of Coordinates If a region in the plane can be describe in polar coordinates as 0 g 1 (θ) r g 2 (θ), α θ β then wee have the following conversion formula f (x, y)da = β g2 (θ) α g 1 (θ) f (rcos(θ), rsin(θ))rdrd θ yan Blair (U Penn) Math 240: Double Integrals and Green s TheoremThursday March 17, / 15

17 Double Integrals Change of Coordinates If a region in the plane can be describe in polar coordinates as 0 g 1 (θ) r g 2 (θ), α θ β then wee have the following conversion formula f (x, y)da = β g2 (θ) α g 1 (θ) f (rcos(θ), rsin(θ))rdrd θ Example Evaluate 3 9 x 2 x2 + y 2 dydx 3 0 yan Blair (U Penn) Math 240: Double Integrals and Green s TheoremThursday March 17, / 15

18 Green s Theorem Green s Theorem Theorem (Green s Theorem) Suppose C is a piecewise smooth simple closed curve bounding a region. If P, Q, P Q and are continuous on, then y x Pdx + Qdy = ( Q x P y )da, C where C is oriented counterclockwise. yan Blair (U Penn) Math 240: Double Integrals and Green s TheoremThursday March 17, / 15