Final answer:
The area of the region between the curves 4xy² = 12 and x = y is 3√3
Step-by-step explanation:
To find the area between the curves 4xy² = 12 and x = y, we need to first determine the points of intersection. By setting the two equations equal to each other, we get 4xy² = 12 = y³. Simplifying, we find y = √3 and y = -√3. The area between the curves can be found by integrating with respect to y from -√3 to √3. The integral is given by:
A = ∫(y-y²/4)dy from -√3 to √3
Solving the integral, we get:
A = [y²/2 - y³/12] from -√3 to √3
Substituting the limits of integration, we have:
A = [3/2 - (3/12 + 3/12)] - [-3/2 + (3/12 + 3/12)]
Simplifying further, we obtain:
A = 3√3