The area of the region bounded by the curves `y=x^(2)andy=(2)/(1+x^(2))` is

Answer

To find the area of the region bounded by the curves \( y = x^2 \) and \( y = \frac{2}{1+x^2} \), we will follow these steps:

### Step 1: Find the points of intersection
To determine the area between the two curves, we first need to find the points where they intersect. This is done by setting the equations equal to each other:

\[ x^2 = \frac{2}{1+x^2} \]

Multiplying both sides by \( 1 + x^2 \) to eliminate the fraction gives:

\[ x^2(1 + x^2) = 2 \]

This simplifies to:

\[ x^4 + x^2 - 2 = 0 \]

Let \( u = x^2 \). Then, we have:

\[ u^2 + u - 2 = 0 \]

### Step 2: Solve the quadratic equation
Using the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ u = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 + 8}}{2} = \frac{-1 \pm 3}{2} \]

This gives us:

\[ u = 1 \quad \text{and} \quad u = -2 \]

Since \( u = x^2 \) must be non-negative, we have \( u = 1 \). Thus, \( x^2 = 1 \) implies \( x = 1 \) or \( x = -1 \).

### Step 3: Set up the integral for the area
The area \( A \) between the curves from \( x = -1 \) to \( x = 1 \) is given by:

\[ A = \int_{-1}^{1} \left( \frac{2}{1+x^2} - x^2 \right) dx \]

### Step 4: Evaluate the integral
We can split the integral:

\[ A = \int_{-1}^{1} \frac{2}{1+x^2} dx - \int_{-1}^{1} x^2 dx \]

The first integral can be computed as:

\[ \int \frac{2}{1+x^2} dx = 2 \tan^{-1}(x) \]

Thus,

\[ \int_{-1}^{1} \frac{2}{1+x^2} dx = 2 \left[ \tan^{-1}(x) \right]_{-1}^{1} = 2 \left( \tan^{-1}(1) - \tan^{-1}(-1) \right) = 2 \left( \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) \right) = 2 \cdot \frac{\pi}{2} = \pi \]

Now, for the second integral:

\[ \int_{-1}^{1} x^2 dx = \left[ \frac{x^3}{3} \right]_{-1}^{1} = \frac{1^3}{3} - \frac{(-1)^3}{3} = \frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3} \]

### Step 5: Combine the results
Now, substituting back into the area formula:

\[ A = \pi - \frac{2}{3} \]

### Final Answer
Thus, the area of the region bounded by the curves is:

\[ A = \pi - \frac{2}{3} \]