Area, Volume, and Center of Mass MATH 311, Calculus III J. obert Buchanan Department of Mathematics Fall 211
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.
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.
Example (1 of 6) Find the area of the triangular region with vertices at (, 2), (3, 2), and (1, 1). 2. 1.5 1..5..5 1. 1.5 2. 2.5 3. x
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 1 1 1 1 2 y 1 dx dy (2y 1) (2 y) dy 3y 3 dy 3 2 y 2 2 3y 1 ( ) 3 6 6 2 3 3 2
Example (3 of 6) Find the area of the bounded region in the xy-plane between y x and y x 2. 1. y.8.6.4.2
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 3 1 1 3
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 1 1 1..5..5 1. 1.5 2. 5 x
Example (6 of 6) A 1 da y y 2 y 1 dx dy y (y 2 y) dy 2y y 2 dy y 2 1 3 y 3 2 4 8 3 4 3
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.
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.
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 2.5 2. 1.5 1...5 y 1. x.5 1..
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
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 8 6 4 2..5 y 1. x.5 1..
Example (4 of 4) V 1 1 1 1 1 1 1 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 3 1 6 dy
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
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 1 2 1 2 3 4 x
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
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
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 128 15 y 2 x(y + 3) dx dy (12 + 4y 3y 2 y 3 ) dy 32
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 128 15 x(y + 3) dx dy y (24 2 + 8y 32 y 4 12 ) y 5 (12 + 4y 3y 2 y 3 ) dy 32 dy 384 5
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 128 15 x(y + 3) dx dy y (24 2 + 8y 32 y 4 12 ) y 5 (x, y) (12 + 4y 3y 2 y 3 ) dy 32 dy 384 5 ( My m, M ) ( x 12 m 5, 4 ) 15
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.
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 1 2 1 2 3 4 x
Example (2 of 2) I x y 2 y 2 (y + 3) dx dy I y y 2 x 2 (y + 3) dx dy
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 4 3 5 y 5 1 6 y 6 2 y 2 x 2 (y + 3) dx dy 128 5
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 4 3 5 y 5 1 6 y 6 2 y 2 x 2 (y + 3) dx dy 64 + 64 3 y y 6 1 3 y 7 dy 64y + 32 3 y 2 1 7 y 7 1 24 y 8 128 5 2 1536 7
Homework ead Section 13.2. Exercises: 1 17 odd, 23 31 odd, 37, 43, 45