The area between the curves `y= x^(2)` and `y = (2)/(1 + x^(2))` is equal to

Options

A. `pi - (1)/(3)` sq. units
B. `pi - 2` sq. units
C. `pi - (2)/(3)` sq. units
D. `pi + (2)/(3)` sq. units

Answer

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

### Step 1: Find the Points of Intersection
We need to set the two equations equal to each other to find the points where they intersect.

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

Multiplying both sides by \( 1 + x^2 \):

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

This simplifies to:

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

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

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

Factoring this quadratic equation:

\[ (u - 1)(u + 2) = 0 \]

Thus, \( u = 1 \) or \( u = -2 \). Since \( u = x^2 \) must be non-negative, we discard \( u = -2 \) and take:

\[ x^2 = 1 \implies x = \pm 1 \]

### Step 2: Set Up the Integral
The area \( A \) between the curves from \( x = -1 \) to \( x = 1 \) can be calculated using the integral:

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

### Step 3: Simplify the Integral
We can simplify the integral as follows:

\[ A = \int_{-1}^{1} \left( \frac{2 - x^2(1 + x^2)}{1 + x^2} \right) \, dx = \int_{-1}^{1} \left( \frac{2 - x^2 - x^4}{1 + x^2} \right) \, dx \]

### Step 4: Calculate the Integral
We can split the integral into two parts:

\[ A = \int_{-1}^{1} \frac{2}{1 + x^2} \, dx - \int_{-1}^{1} \frac{x^2 + x^4}{1 + x^2} \, dx \]

The first integral is:

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

The second integral can be calculated as follows:

\[ \int_{-1}^{1} x^2 \, dx = 2 \cdot \frac{x^3}{3} \bigg|_{0}^{1} = \frac{2}{3} \]

And for \( x^4 \):

\[ \int_{-1}^{1} x^4 \, dx = 2 \cdot \frac{x^5}{5} \bigg|_{0}^{1} = \frac{2}{5} \]

Thus, we have:

\[ \int_{-1}^{1} (x^2 + x^4) \, dx = \frac{2}{3} + \frac{2}{5} = \frac{10 + 6}{15} = \frac{16}{15} \]

### Step 5: Combine the Results
Now we can combine the results:

\[ A = \frac{\pi}{2} - \frac{16}{15} \]

### Final Result
Thus, the area between the curves is:

\[ A = \frac{\pi}{2} - \frac{16}{15} \]