Final answer:
The double integral ∫∫_D (2x+y) dA over the domain D is solved using iterated integrals, starting with integration with respect to x and followed by y, using the limits provided by the domain's inequalities.
Step-by-step explanation:
The student has asked to evaluate the double integral of the function 2x+y over the domain D, given by the inequalities 1 ≤ y ≤ 4 and y − 3 ≤ x ≤ 3.
To solve this, we will apply the iterated integral technique, integrating with respect to x first, and then with respect to y.
First, we set up the integral with respect to x: ∫ (2x+y) dx from x = y - 3 to x = 3.
Next, we compute the antiderivative with respect to x, which will be x² + yx, evaluated from y - 3 to 3.
After simplifying the expression, we then set up the integral with respect to y from 1 to 4.
We compute the antiderivative with respect to y and then evaluate it at the limits of integration to obtain the final result.