Question

Find the mass of the lamina described by the inequalities, given that its density is rho(x, y) = xy. 0 ≤ x ≤ 2, 0 ≤ y ≤ 2

Answer 1

Answer: Mass of lamina = 4

Step-by-step explanation: A lamina is a plate in 2 dimensions, described by the plane it covers and its density function, $\rho(x,y)$.

To determine mass of the lamina:

mass (M) = $\int {\int\limits_D \rho(x,y) \, dA$

where D is region bounded by the axis.

For the question:

M = $\int\limits^2_0 {\int\limits^2_0 xy \, dy \,dx$

Calculating the double integral:

M = $\int\limits^2_0 { x\frac{y^{2}}{2} \,dx$

M = $\int\limits^2_0 { x(\frac{2^{2}}{2}-0)} \,dx$

M = $\int\limits^2_0 { 2x} \,dx$

M = $\frac{2.2^{2}}{2} - 0$

M = 4

The mass of lamina is 4 units.