Question

Find the mass of the lamina that occupies the region D = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} with the density function ρ(x, y) = xye^(x+y)

Answer 1

Answer:

The mass of the lamina is 1

Step-by-step explanation:

Let $\rho(x,y)$ be a continuous density function of a lamina in the plane region D,then the mass of the lamina is given by:

$m=\int\limits \int\limits_D \rho(x,y) \, dA$.

From the question, the given density function is $\rho (x,y)=xye^{x+y}$.

Again, the lamina occupies a rectangular region: D={(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.

The mass of the lamina can be found by evaluating the double integral:

$I=\int\limits^1_0\int\limits^1_0xye^{x+y}dydx$.

Since D is a rectangular region, we can apply Fubini's Theorem to get:

$I=\int\limits^1_0(\int\limits^1_0xye^{x+y}dy)dx$.

Let the inner integral be: $I_0=\int\limits^1_0xye^{x+y}dy$, then

$I=\int\limits^1_0(I_0)dx$.

The inner integral is evaluated using integration by parts.

Let $u=xy$, the partial derivative of u wrt y is

$\implies du=xdy$

and

$dv=\int\limits e^{x+y} dy$, integrating wrt y, we obtain

$v=\int\limits e^{x+y}$

Recall the integration by parts formula:$\int\limits udv=uv- \int\limits vdu$

This implies that:

$\int\limits xye^{x+y}dy=xye^{x+y}-\int\limits e^{x+y}\cdot xdy$

$\int\limits xye^{x+y}dy=xye^{x+y}-xe^{x+y}$

$I_0=\int\limits^1_0 xye^{x+y}dy$

We substitute the limits of integration and evaluate to get:

$I_0=xe^x$

This implies that:

$I=\int\limits^1_0(xe^x)dx$.

Or

$I=\int\limits^1_0xe^xdx$.

We again apply integration by parts formula to get:

$\int\limits xe^xdx=e^x(x-1)$.

$I=\int\limits^1_0xe^xdx=e^1(1-1)-e^0(0-1)$.

$I=\int\limits^1_0xe^xdx=0-1(0-1)$.

$I=\int\limits^1_0xe^xdx=0-1(-1)=1$.

No unit is given, therefore the mass of the lamina is 1.