The area of the closed figure bounded by the curves
`y=cosx,y =1+(2)/(pi)x and x=pi//2,` is

Options

A. `(pi+4)/(4)`
B. `(3pi-4)/(4)`
C. `(3pi)/(4)`
D. `(pi)/(4)`

Answer

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

### Step 1: Identify the curves and the region of integration
We have two curves:
1. \( y = \cos x \)
2. \( y = 1 + \frac{2}{\pi} x \)

The vertical line \( x = \frac{\pi}{2} \) serves as one boundary of the region. We need to find the area between these two curves from \( x = 0 \) to \( x = \frac{\pi}{2} \).

### Step 2: Set up the integral for the area
The area \( A \) between the curves can be calculated using the formula:
\[ A = \int_{x_1}^{x_2} (y_2 - y_1) \, dx \]
where \( y_2 \) is the upper curve and \( y_1 \) is the lower curve.

In this case:
- \( y_1 = \cos x \) (the lower curve)
- \( y_2 = 1 + \frac{2}{\pi} x \) (the upper curve)

Thus, the area can be expressed as:
\[ A = \int_{0}^{\frac{\pi}{2}} \left( \left(1 + \frac{2}{\pi} x\right) - \cos x \right) \, dx \]

### Step 3: Simplify the integral
Now, we simplify the expression inside the integral:
\[ A = \int_{0}^{\frac{\pi}{2}} \left( 1 + \frac{2}{\pi} x - \cos x \right) \, dx \]

### Step 4: Evaluate the integral
We can split the integral into three parts:
\[ A = \int_{0}^{\frac{\pi}{2}} 1 \, dx + \int_{0}^{\frac{\pi}{2}} \frac{2}{\pi} x \, dx - \int_{0}^{\frac{\pi}{2}} \cos x \, dx \]

Calculating each part:
1. \( \int_{0}^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} \)
2. \( \int_{0}^{\frac{\pi}{2}} \frac{2}{\pi} x \, dx = \frac{2}{\pi} \left[ \frac{x^2}{2} \right]_{0}^{\frac{\pi}{2}} = \frac{2}{\pi} \cdot \frac{\left(\frac{\pi}{2}\right)^2}{2} = \frac{2}{\pi} \cdot \frac{\pi^2}{8} = \frac{\pi}{4} \)
3. \( \int_{0}^{\frac{\pi}{2}} \cos x \, dx = \left[ \sin x \right]_{0}^{\frac{\pi}{2}} = 1 - 0 = 1 \)

Putting it all together:
\[ A = \frac{\pi}{2} + \frac{\pi}{4} - 1 \]

### Step 5: Combine the results
To combine \( \frac{\pi}{2} \) and \( \frac{\pi}{4} \):
\[ \frac{\pi}{2} = \frac{2\pi}{4} \]
Thus,
\[ A = \frac{2\pi}{4} + \frac{\pi}{4} - 1 = \frac{3\pi}{4} - 1 \]

### Step 6: Final result
The area of the closed figure bounded by the curves is:
\[ A = \frac{3\pi - 4}{4} \]