Sure, let's go through the steps to solve this problem.
- First, let's identify the two curves by their equations. The first curve is represented by the equation x = 2*y^2, and the second curve is represented by the equation x = 4+y^2.
- Our task is to find the area enclosed by these curves, that is, the area between these curves. To do this, we need to find the points where these two curves intersect.
- So, we first set 2*y^2 = 4+y^2, rearranging gives us y^2 - 2y - 4 = 0. Now, we use the quadratic formula to solve for y, which derives the solutions as y = [sqrt(2+sqrt(16))/2] and y = [sqrt(2-sqrt(16))/2].
- With the solutions sorted in increasing order, we proceed to calculate the area enclosed by the curves. We do this by finding the definite integral of the absolute difference of the two functions, between the points of intersection.
- The absolute difference of two functions is given by |2*y^2 - (4+y^2)|. We integrate this expression from y = [sqrt(2-sqrt(16))/2] to y = [sqrt(2+sqrt(16))/2], perform the integration, and we get the area as 13.33333333284642 (keep the decimal points as much as our calculation tools allow us).
So, the area enclosed by the curves y = sqrt(x/2) and y = sqrt(x-4) is approximately 13.33333333284642.