Math Applications of Double Integrals

Transcription

1 Math Applications of Double Integrals Peter A. Perry University of Kentucky October 26, 2018

2 Homework Re-read section 15.4 Begin work on problems 1-23 (odd) from 15.4 Read section 15.5 for Monday, October 29 Finish up Webwork C1

3 Unit III: Multiple Integrals Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture 29 Lecture 30 Lecture 31 Lecture 32 Lecture 33 Lecture 34 Double Integrals over Rectangles Double Integrals over General Regions Double Integrals in Polar Coodinates Applications of Double Integrals Surface Area Triple Integrals Triple Integrals in Cylindrical Coordinates Triple Integrals in Spherical Coordinates Change of Variable in Multiple Integrals, Part I Change of Variable in Multiple Integrals, Part II Exam III Review

4 Goals of the Day Compute Mass of a Lamina from Mass Density Compute Moments and Centers of Mass Compute Moments of Inertia

5 Density to Mass The density of a plane solid (lamina) D is the mass per unit area: (x, y ) ρ(x, y) = m lim A 0 A. The total mass is approximately M n ρ(x, y ) A i,j=1 where (i, j) index rectangles in D of area A The total mass is exactly M = ρ(x, y) da D

6 Charge Density to Charge The charge density of a plane conductor D is the mass per unit area: (x, y ) ρ(x, y) = Q lim A 0 A. The total mass is approximately Q n ρ(x, y ) A i,j=1 where (i, j) index rectangles in D of area A The total mass is exactly Q = ρ(x, y) da D

7 Density to Mass Electric charge is distributed over the disc x 2 + y 2 1m with charge density σ(x, y) = x 2 + y 2 coulombs per square meter. Find the total charge Q on the disc.

8 Moments, Centers of Mass - Discrete Systems For a set of point masses m i at (x i, y i ): y The total mass is M = n i=1 m i m 2 (x 2, y 2 ) m 3 (x 3, y 3 ) m 1 (x 1, y 1 ) m 4 (x 4, y 4 ) x

9 Moments, Centers of Mass - Discrete Systems For a set of point masses m i at (x i, y i ): y The total mass is M = n i=1 m i The moment with respect to the x-axis is M x = n i=1 y i m i m 2 (x 2, y 2 ) m 3 (x 3, y 3 ) m 1 (x 1, y 1 ) m 4 (x 4, y 4 ) x

10 Moments, Centers of Mass - Discrete Systems For a set of point masses m i at (x i, y i ): y m 2 (x 2, y 2 ) m 3 (x 3, y 3 ) m 1 (x 1, y 1 ) m 4 (x 4, y 4 ) x The total mass is M = n i=1 m i The moment with respect to the x-axis is M x = n i=1 y i m i The moment with respect to the y-axis is M y = n i=1 x i m i

11 Moments, Centers of Mass - Discrete Systems For a set of point masses m i at (x i, y i ): The total mass is M = n i=1 m i y The moment with respect to the x-axis is M x = n i=1 y i m i m 2 (x 2, y 2 ) m 3 (x 3, y 3 ) m 1 (x 1, y 1 ) m 4 (x 4, y 4 ) x The moment with respect to the y-axis is M y = n i=1 x i m i The center of mass is ( My (x, y) = M, M ) x M

12 Moments, Centers of Mass - Discrete Systems For a set of point masses m i at (x i, y i ): The total mass is M = n i=1 m i y The moment with respect to the x-axis is M x = n i=1 y i m i m 2 (x 2, y 2 ) m 3 (x 3, y 3 ) m 1 (x 1, y 1 ) m 4 (x 4, y 4 ) x The moment with respect to the y-axis is M y = n i=1 x i m i The center of mass is ( My (x, y) = M, M ) x M ( n (x, y) = i=1 x i m i n i=1 m i, n i=1 y ) i m i n i=1 m i

13 Moments, Centers of Mass - Continuous Systems (x, y ) If a lamina D has density ρ(x, y): M ρ(x, y ) A M x y ρ(x, y ) A M y x ρ(x, y ) A Taking limits...

14 Moments, Centers of Mass - Continuous Systems (x, y ) If a lamina D has density ρ(x, y): M ρ(x, y ) A M x y ρ(x, y ) A M y x ρ(x, y ) A Taking limits... M = ρ(x, y) da D M x = yρ(x, y) da D M y = xρ(x, y) da D

15 Center of Mass M ρ(x, y ) A M x y ρ(x, y ) A M y x ρ(x, y ) A A lamina occupies the region D = {(x, y) : x 2 + y 2 1, x 0, y 0}. Find its center of mass if the density at each point is proportional to its distance from the x-axis.

16 Moment of Inertia - Discrete Systems For a set of point masses m i at (x i, y i ): y m 2 (x 2, y 2 ) m 3 (x 3, y 3 ) m 1 (x 1, y 1 ) m 4 (x 4, y 4 ) x

17 Moment of Inertia - Discrete Systems For a set of point masses m i at (x i, y i ): y m 2 (x 2, y 2 ) m 3 (x 3, y 3 ) m 1 (x 1, y 1 ) m 4 (x 4, y 4 ) x The moment of inertia about the x-axis is I x = n m i yi 2 i=1

18 Moment of Inertia - Discrete Systems For a set of point masses m i at (x i, y i ): The moment of inertia about the x-axis is y I x = n m i yi 2 i=1 m 1 m 2 (x 2, y 2 ) m 3 (x 3, y 3 ) (x 1, y 1 ) m 4 (x 4, y 4 ) x The moment of inertia about the y-axis is I y = n m i xi 2 i=1

19 Moment of Inertia - Discrete Systems For a set of point masses m i at (x i, y i ): The moment of inertia about the x-axis is y I x = n m i yi 2 i=1 m 1 m 2 (x 2, y 2 ) m 3 (x 3, y 3 ) (x 1, y 1 ) m 4 (x 4, y 4 ) x The moment of inertia about the y-axis is I y = n m i xi 2 i=1 The moment of intertia about the origin is I 0 = n m i (xi 2 + yi 2) i=1

20 Moment of Inertia - Continuous Systems (x, y ) If a lamina D has density ρ(x, y): I x (y )2 ρ(x, y ) A I y (x )2 ρ(x, y ) A M y ( (x )2 + (y )2) ρ(x, y ) A Taking limits...

21 Moment of Inertia - Continuous Systems (x, y ) If a lamina D has density ρ(x, y): I x (y )2 ρ(x, y ) A I y (x )2 ρ(x, y ) A M y ( (x )2 + (y )2) ρ(x, y ) A Taking limits... I x = y 2 ρ(x, y) da D I y = x 2 ρ(x, y) da D I 0 = (x 2 + y 2 )ρ(x, y) da D

22 Moments of Inertia I x = y 2 ρ(x, y) da D I y = x 2 ρ(x, y) da D I 0 = (x 2 + y 2 )ρ(x, y) da D a Find the moments of inertia I x, I y, and I 0 for the part of the disc of radius a in the first quadrant, assuming that the density is constant.