Find the area of the region enclosed by the given curves .
`y=cos x , y=1 -(2x)/(pi)`

Answer

To find the area of the region enclosed by the curves \( y = \cos x \) and \( y = 1 - \frac{2x}{\pi} \), we will follow these steps:

### Step 1: Find the points of intersection
We need to find the points where the two curves intersect. This is done by setting the equations equal to each other:

\[ \cos x = 1 - \frac{2x}{\pi} \]

To find the points of intersection, we can evaluate the curves at specific values of \( x \):

- For \( x = 0 \):
\[ y = \cos(0) = 1 \quad \text{and} \quad y = 1 - \frac{2(0)}{\pi} = 1 \]
So, the point \( (0, 1) \) is an intersection point.

- For \( x = \frac{\pi}{2} \):
\[ y = \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad y = 1 - \frac{2\left(\frac{\pi}{2}\right)}{\pi} = 0 \]
So, the point \( \left(\frac{\pi}{2}, 0\right) \) is another intersection point.

### Step 2: Set up the integral for the area
The area \( A \) between the curves from \( x = 0 \) to \( x = \frac{\pi}{2} \) can be calculated using the integral:

\[ A = \int_{0}^{\frac{\pi}{2}} \left( \cos x - \left(1 - \frac{2x}{\pi}\right) \right) \, dx \]

This simplifies to:

\[ A = \int_{0}^{\frac{\pi}{2}} \left( \cos x - 1 + \frac{2x}{\pi} \right) \, dx \]

### Step 3: Calculate the integral
Now we can compute the integral:

\[ A = \int_{0}^{\frac{\pi}{2}} \cos x \, dx - \int_{0}^{\frac{\pi}{2}} 1 \, dx + \int_{0}^{\frac{\pi}{2}} \frac{2x}{\pi} \, dx \]

Calculating each part:

1. **First integral**:
\[ \int_{0}^{\frac{\pi}{2}} \cos x \, dx = \left[ \sin x \right]_{0}^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1 \]

2. **Second integral**:
\[ \int_{0}^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]

3. **Third integral**:
\[ \int_{0}^{\frac{\pi}{2}} \frac{2x}{\pi} \, dx = \frac{2}{\pi} \left[ \frac{x^2}{2} \right]_{0}^{\frac{\pi}{2}} = \frac{2}{\pi} \cdot \frac{(\frac{\pi}{2})^2}{2} = \frac{2}{\pi} \cdot \frac{\pi^2}{8} = \frac{\pi}{4} \]

### Step 4: Combine the results
Now, substituting back into the area formula:

\[ A = 1 - \frac{\pi}{2} + \frac{\pi}{4} \]

To combine these, convert \( 1 \) into a fraction with a common denominator of \( 4 \):

\[ A = \frac{4}{4} - \frac{2\pi}{4} + \frac{\pi}{4} = \frac{4 - 2\pi + \pi}{4} = \frac{4 - \pi}{4} \]

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

\[ \boxed{\frac{4 - \pi}{4}} \text{ square units} \]