Final answer:
To find the area of the region between the curves y=x² and y=2/(x²+1), we need to find the points of intersection and evaluate the definite integral.
Step-by-step explanation:
To find the area of the region between the curves y=x² and y=2/(x²+1), we need to find the points of intersection. Set the two equations equal to each other and solve for x: x² = 2/(x²+1). Multiply both sides by x²+1 to get x^4 + x² - 2 = 0. Factor the quadratic equation as (x²-1)(x²+2) = 0. This gives us x = 1 and x = -1 as the points of intersection.
To find the area, we integrate the difference between the two curves from x = -1 to x = 1. The integral of y=x² is (1/3)x³+C, and the integral of y=2/(x²+1) is 2arctan(x)+C. Evaluating the integrals at the limits of integration, we get (1/3)(1)³ - (1/3)(-1)³ + 2arctan(1) - 2arctan(-1) = 2/3 + 2π.
Therefore, the area of the region between the curves y=x² and y=2/(x²+1) is 2/3 + 2π units squared.