Area, Volume, and Center of Mass

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1 Area, Volume, and Center of Mass MATH 311, Calculus III J. obert Buchanan Department of Mathematics Fall 211

2 Introduction Today we will apply the double integral to the problems of finding the area of regions in the plane, the volume of a bounded region in three dimensions, the center of mass of a region in the plane.

3 Area If is a region in the xy-plane, the area A of is A 1 da provided the value of the double integral is finite.

4 Example (1 of 6) Find the area of the triangular region with vertices at (, 2), (3, 2), and (1, 1) x

5 Example (2 of 6) The region is bounded by the graphs of the lines x 2 y and x 2y 1 for 1 y 2. A 1 da y y 1 dx dy (2y 1) (2 y) dy 3y 3 dy 3 2 y 2 2 3y 1 ( )

6 Example (3 of 6) Find the area of the bounded region in the xy-plane between y x and y x y

7 Example (4 of 6) A 1 da 1 x 1 x 2 1 dy dx x x 2 dx 2 3 x 3/2 1 3 x

8 Example (5 of 6) Find the area of the bounded region in the xy-plane between y x and x y 2 y. 2 y x

9 Example (6 of 6) A 1 da y y 2 y 1 dx dy y (y 2 y) dy 2y y 2 dy y y

10 Volume of a Solid egion If f (x, y) for all (x, y) in a region, the volume of the solid below the surface z f (x, y) and above the region in the xy-plane is V f (x, y) da.

11 Volume of a Solid egion If f (x, y) for all (x, y) in a region, the volume of the solid below the surface z f (x, y) and above the region in the xy-plane is V f (x, y) da. If f (x, y) g(x, y) for all (x, y) in a region, the volume of the solid below the surface z f (x, y) and above the surface z g(x, y) within the the region in the xy-plane is V (f (x, y) g(x, y)) da.

12 Example (1 of 4) Find the volume of the solid region below the graph of z e y 2 and above the triangle in the xy-plane with vertices at (, ), (1, 1), (, 1). z y 1. x.5 1..

13 Example (2 of 4) V 1 y 1 1 f (x, y) da e y 2 da xe y 2 y 1 2 ey 2 1 ye y 2 dy 1 2 e 1 2 e y 2 dx dy dy

14 Example (3 of 4) Find the volume of the solid region above the graph of z 2x 2 + y 2 and below the graph of z 8 x 2 2y 2 and within the rectangle {(x, y) x 1, y 1}. z y 1. x.5 1..

15 Example (4 of 4) V f (x, y) da (8 x 2 2y 2 ) (2x 2 + y 2 ) dx dy (8 3x 2 3y 2 ) dx dy 8x x 3 3xy 2 1 (8 1 3y 2 ) dy (7 3y 2 ) dy 7y y dy

16 Moments and Center of Mass lamina: a thin sheet of material in the shape of a region 2. ρ(x, y): the density of the lamina at (x, y). m: mass of the lamina m ρ(x, y) da. M x : moment with respect to the x-axis M x yρ(x, y) da. M y : moment with respect to the y-axis M y xρ(x, y) da. (x, y) center of mass (x, y) ( My m, M ) x m

17 Example (1 of 2) A lamina with density ρ(x, y) y + 3 is bounded by the curves x y 2 and x 4. Find its mass, moments, and center of mass. 1 y x

18 Example (2 of 2) m M x y 2 (y + 3) dx dy y 2 y(y + 3) dx dy M y y 2 x(y + 3) dx dy

19 Example (2 of 2) m M x y 2 (y + 3) dx dy y 2 y(y + 3) dx dy (12 + 4y 3y 2 y 3 ) dy 32 M y y 2 x(y + 3) dx dy

20 Example (2 of 2) m M x M y y 2 (y + 3) dx dy y 2 y(y + 3) dx dy (12y + 4y 2 3y 3 y 4 ) dy y 2 x(y + 3) dx dy (12 + 4y 3y 2 y 3 ) dy 32

21 Example (2 of 2) m M x M y y 2 (y + 3) dx dy y 2 y(y + 3) dx dy (12y + 4y 2 3y 3 y 4 ) dy x(y + 3) dx dy y ( y 32 y 4 12 ) y 5 (12 + 4y 3y 2 y 3 ) dy 32 dy 384 5

22 Example (2 of 2) m M x M y y 2 (y + 3) dx dy y 2 y(y + 3) dx dy (12y + 4y 2 3y 3 y 4 ) dy x(y + 3) dx dy y ( y 32 y 4 12 ) y 5 (x, y) (12 + 4y 3y 2 y 3 ) dy 32 dy ( My m, M ) ( x 12 m 5, 4 ) 15

23 Moment of Inertia Inertia: tendency of an object to resist a change in motion. I x : moment of inertia about the x-axis I x y 2 ρ(x, y) da. I y : moment of inertia about the y-axis I y x 2 ρ(x, y) da.

24 Example (1 of 2) A lamina with density ρ(x, y) y + 3 is bounded by the curves x y 2 and x 4. Find its moments of inertia. 1 y x

25 Example (2 of 2) I x y 2 y 2 (y + 3) dx dy I y y 2 x 2 (y + 3) dx dy

26 Example (2 of 2) I x I y y 2 y 2 (y + 3) dx dy 12y 2 + 4y 3 3y 4 y 5 dy 4y 3 + y y y 6 2 y 2 x 2 (y + 3) dx dy 128 5

27 Example (2 of 2) I x I y y 2 y 2 (y + 3) dx dy 12y 2 + 4y 3 3y 4 y 5 dy 4y 3 + y y y 6 2 y 2 x 2 (y + 3) dx dy y y y 7 dy 64y y y y

28 Homework ead Section Exercises: 1 17 odd, odd, 37, 43, 45

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