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.
Find the mass of the rectangular region 0≤x≤1
, 0≤y≤2
with density function ρ(x,y)=2−y
.
1 answer