Find center of mass of the lamina that occupies the region D and has the given density function ρ.
D is the triangular region with vertices (0, 0), (2, 1), (0, 3); ρ(x, y) = 5(x + y)
mass=30
Find center of mass
To find the center of mass of the lamina occupying the given region D with density function ρ(x, y) = 5(x + y), we can use the following steps:
Step 1: Find the mass of the lamina.
Given the density function ρ(x, y) = 5(x + y), we can calculate the mass of the lamina by integrating the density function over the region D.
The formula for the mass is given by:
M = ∫∫D ρ(x, y) dA
Since the density function ρ(x, y) = 5(x + y) and the region D is a triangular region, we can set up the integral as follows:
M = ∫∫D 5(x + y) dA
Step 2: Evaluate the integral to find the mass.
To evaluate the integral, we need to express the region D in terms of suitable limits of integration. In this case, we can express the region D as follows:
D: 0 ≤ x ≤ 2, 0 ≤ y ≤ (3 - 2x/2)
Therefore, the integral becomes:
M = ∫(0 to 2) ∫(0 to 3 - 2x/2) 5(x + y) dy dx
Evaluating this double integral will give us the mass of the lamina, which is given as 30 in this case.
Step 3: Find the moments about the x and y-axes.
The moments about the x and y-axes are given by the following formulas:
Mx = ∫∫D yρ(x, y) dA
My = ∫∫D xρ(x, y) dA
Using the density function ρ(x, y) = 5(x + y), we can set up the integrals as follows:
Mx = ∫∫D y(5(x + y)) dA
My = ∫∫D x(5(x + y)) dA
Step 4: Evaluate the integrals to find the moments.
Using the limits of integration for the region D, we can evaluate the integrals for Mx and My to find the moments about the x and y-axes.
Step 5: Find the coordinates of the center of mass.
The coordinates of the center of mass (x̄, ȳ) are given by the following formulas:
x̄ = My / M
ȳ = Mx / M
Dividing the moments about the x and y-axes by the mass of the lamina will give us the coordinates of the center of mass.
By following these steps, you should be able to find the center of mass of the given lamina occupying the region D.