Final answer:
To find the area enclosed by the curves y=x⁴-2x²+4 and y=2x², we need to find the points of intersection and evaluate the integral of the difference of the two curves.
Step-by-step explanation:
To find the area enclosed by the curves y=x⁴-2x²+4 and y=2x², we need to find the points of intersection. Setting the two equations equal to each other, we get x⁴-2x²+4 = 2x². Simplifying, we get x⁴-4x²+4 = 0. Factoring this as a quadratic equation, we get (x²-2)² = 0. Solving for x, we find two solutions: x = ±√2.
Next, we integrate the difference of the two curves from x = -√2 to x = √2. The area is given by the integral of (2x² - (x⁴-2x²+4)) dx, evaluated from -√2 to √2. Simplifying and evaluating the integral, we get A = 2√2 - (8√2/3 - 8√2/5 + 32/15).