A lamina with density $\delta$(x, y) = 2x² + y² + 9 is bounded by x = y, x = 0, y = 0, y = 2. Find its center of mass.

Question

Quizplus · Accepted Answer

The center of mass of the lamina is given by the formulas:

Xcm = (1/M) * ∫∫(xD)dA,    Ycm = (1/M) * ∫∫(yD)dA,

where M is the total mass of the lamina, D is the distance function from the point (x, y) to the y-axis, and (x, y) is a point in the lamina.

We first need to find the total mass of the lamina, which is given by:

M = ∫∫ρ(x, y)dA,

where ρ(x, y) is the density function.

M = ∫0^2 ∫0^x (2x + y + 9) dxdy
M = ∫0^2 [x^2 + xy + 9x]dy
M = ∫0^2 (2x^3 + 27x)dx
M = (1/2)(2)(2^4) + (1/2)(27)(2^2)
M = 37

Next, we need to evaluate the integrals for Xcm and Ycm:

Xcm = (1/37) * ∫0^2 ∫0^x (xD)dA
Xcm = (1/37) * ∫0^2 [x^3/3 + xy^2/2 + 9xy]dy
Xcm = (1/37) * ∫0^2 (2x^4/12 + 9x^2/2)
Xcm = (1/37) * [(2/60)(2^5) + (9/2)(2)]
Xcm = 8/37

Ycm = (1/37) * ∫0^2 ∫0^x (yD)dA
Ycm = (1/37) * ∫0^2 [x^2y + y^2/2 + 9y]dy
Ycm = (1/37) * ∫0^2 (2x^2 + 5/2 + 18)
Ycm = (1/37) * (2(2^3) + (5/2)(2) + 18(2))
Ycm = 49/37

Therefore, the center of mass of the lamina is (8/37, 49/37), which corresponds to choice (B).

A Lamina with Density δ\deltaδ (X, Y) = 2x2 + Y2 + 9 Is Bounded

A Lamina with Density $\delta$ (X, Y) = 2x² + Y² + 9 Is Bounded