Question

Compute the double integral over the region D where D is the region bounded by x = 2y, y = -x, and y = 2:

∬[D] (dA / (y^2 + 1))

Answer 1

To compute the double integral over the given region D, you need to find the limits of integration for both x and y. The region D is bounded by the lines x = 2y, y = -x, and y = 2.

Step-by-step explanation:

To compute the double integral over the region D, we need to find the limits of integration for both x and y. The region D is bounded by the lines x = 2y, y = -x, and y = 2. To find the limits of integration for x, we solve the equation x = 2y for y, which gives us y = x/2. Since y is bounded above by y = 2, the limits of integration for x are from x/2 to 2. To find the limits of integration for y, we solve the equation y = -x for x, which gives us x = -y. Since y is bounded above by y = 2, the limits of integration for y are from -2 to 2. Therefore, the double integral over the region D is given by:

∬[D] (dA / (y^2 + 1)) = ∫-22 ∫x/22 (1 / (y^2 + 1)) dx dy.