What is the total area bounded by the curve

y^2(1-x) = x^2(1+x) and the line x = 1

To find the total area bounded by the curve y^2(1-x) = x^2(1+x) and the line x = 1, we can use the method of integration.

The given equation y^2(1-x) = x^2(1+x) can be rearranged to isolate y in terms of x:

y = sqrt[(x^2(1+x))/(1-x)]

Now, we need to find the points of intersection between the curve and the line x = 1.

When x = 1, the equation becomes:

y = sqrt[(1^2(1+1))/(1-1)] = sqrt[2]

So, the curve intersects the line x = 1 at the point (1, sqrt[2]).

Now, let's find the points of intersection between the curve and the x-axis:

When y = 0, the equation becomes:

0 = sqrt[(x^2(1+x))/(1-x)]

To solve this equation, we square both sides:

0 = (x^2(1+x))/(1-x)

Multiplying both sides by (1-x), we get:

0 = x^2(1+x)

This equation is satisfied when x = 0 and x = -1.

So, the curve intersects the x-axis at the points (0, 0) and (-1, 0).

Now, we can set up the integral to find the area bounded by the curve and the line x = 1.

Since the curve is symmetric about the y-axis, we can calculate the area for x between -1 and 1, and then multiply it by 2 to account for the other side.

The integral to calculate the area is:

A = 2 * ∫[from -1 to 1] y dx

To express y in terms of x, recall that y = sqrt[(x^2(1+x))/(1-x)].

So, the integral becomes:

A = 2 * ∫[from -1 to 1] sqrt[(x^2(1+x))/(1-x)] dx

Unfortunately, this integral cannot be solved analytically with simple functions. We would need to use numerical methods or software to evaluate this integral and find the total area bounded by the curve y^2(1-x) = x^2(1+x) and the line x = 1.