Find the mass of the rectangular region 0≤x≤1
, 0≤y≤2
with density function ρ(x,y)=2−y
.
To find the mass of the rectangular region, we need to integrate the density function ρ(x,y) over the given region.
The mass M is given by the double integral of ρ(x,y) over the region:
M = ∬ ρ(x,y) dA
where dA represents an infinitesimal area element.
In this case, the rectangular region is defined by 0≤x≤1 and 0≤y≤2. Therefore, the mass M is given by:
M = ∫[0,1] ∫[0,2] (2-y) dy dx
We can integrate with respect to y first, from 0 to 2:
M = ∫[0,1] [(2y - (1/2)y^2)]|[0,2] dx
M = ∫[0,1] [(4 - 2)] dx
M = ∫[0,1] 2 dx
M = 2x |[0,1]
M = 2(1) - 2(0)
M = 2
So, the mass of the rectangular region 0≤x≤1, 0≤y≤2 with density function ρ(x,y)=2−y is 2.
To find the mass of the rectangular region, we need to integrate the density function over the given region.
First, let's set up the integral:
M = ∬ρ(x,y) dA
We can rewrite the density function ρ(x,y) = 2 - y as ρ(x,y) = 2 - y^1.
Now, let's integrate ρ(x,y) over the given rectangular region 0 ≤ x ≤ 1, 0 ≤ y ≤ 2:
M = ∫(0 to 1) ∫(0 to 2) (2 - y) dy dx
Performing the inner integral:
M = ∫(0 to 1) [2y - 0.5y^2] dy
M = [(y^2) - (0.5y^3/3)] evaluated from y = 0 to y = 2
M = [(2^2) - (0.5(2^3)/3)] - [(0^2) - (0.5(0^3)/3)]
M = [4 - (8/3)] - [0 - 0]
M = 4 - (8/3)
M = (12/3) - (8/3)
M = 4/3
Therefore, the mass of the rectangular region 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 with the density function ρ(x,y) = 2 - y is 4/3.