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)

The mass of the lamina is 1

Step-by-step explanation:

Let be a continuous density function of a lamina in the plane region D,then the mass of the lamina is given by:

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From the question, the given density function is .

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:

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

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Let the inner integral be: , then

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The inner integral is evaluated using integration by parts.

Let , the partial derivative of u wrt y is

and

, integrating wrt y, we obtain

Recall the integration by parts formula:

This implies that:

We substitute the limits of integration and evaluate to get:

This implies that:

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Or

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We again apply integration by parts formula to get:

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.

.

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