Final answer:
To find the area of the region between the curves 4xy² = 12 and x = y, we can set up a double integral. The limits of integration are y = 0 and y = ∛3. The area of the region is 3/2.
Step-by-step explanation:
To find the area of the region between the curves 4xy² = 12 and x = y, we can set up a double integral. We need to find the limits of integration first. From the equation x = y, we can substitute y for x in the other equation:
4y(y^2) = 12
Simplifying, we get:
4y^3 = 12
y^3 = 3
From this equation, we can see that y = ∛3 is the upper limit of integration. The lower limit is y = 0 since the region is bounded by the x-axis. The integral to find the area is:
∫[0,∛3] ∫[y,3/y²] dx dy
This integral can be evaluated to find the area of the region, and the answer is option A: 3/2.