The area of the figure bounded by right of the line `y = x + 1, y= cos x` and x-axis is :

Options

A. `1/2`
B. `2/3`
C. `5/6`
D. `3/2`

Answer

To find the area of the figure bounded by the line \( y = x + 1 \), the curve \( y = \cos x \), and the x-axis, we will follow these steps:

### Step 1: Identify the Points of Intersection
First, we need to find the points where the line intersects the curve \( y = \cos x \).

Set the equations equal to each other:
\[ x + 1 = \cos x \]
This equation does not have a straightforward algebraic solution, so we will find the intersection points graphically or numerically.

### Step 2: Determine the Bounds of Integration
From the graph, we can see that the line intersects the x-axis at \( (-1, 0) \) and the y-axis at \( (0, 1) \). The curve \( y = \cos x \) will intersect the line \( y = x + 1 \) at some point between \( x = 0 \) and \( x = \frac{\pi}{2} \).

### Step 3: Calculate the Area
The area \( A \) can be calculated as the sum of the area of the triangle formed by the points \( O(0, 0) \), \( A(0, 1) \), and \( B(-1, 0) \), and the area under the curve \( y = \cos x \) from \( x = 0 \) to \( x = \frac{\pi}{2} \).

#### Area of Triangle \( OAB \)
The area of triangle \( OAB \) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is 1 (from \( O(0, 0) \) to \( A(0, 1) \)) and the height is also 1 (from \( O(0, 0) \) to \( B(-1, 0) \)):
\[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \]

#### Area Under the Curve \( y = \cos x \)
Next, we calculate the area under the curve from \( x = 0 \) to \( x = \frac{\pi}{2} \):
\[ \text{Area}_{\text{curve}} = \int_0^{\frac{\pi}{2}} \cos x \, dx \]
Calculating the integral:
\[ \int \cos x \, dx = \sin x \]
Evaluating the definite integral:
\[ \text{Area}_{\text{curve}} = \left[ \sin x \right]_0^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1 \]

### Step 4: Total Area
Now, we combine the areas:
\[ \text{Total Area} = \text{Area}_{\text{triangle}} + \text{Area}_{\text{curve}} = \frac{1}{2} + 1 = \frac{3}{2} \]

### Final Answer
Thus, the area of the figure bounded by the line \( y = x + 1 \), the curve \( y = \cos x \), and the x-axis is:
\[ \boxed{\frac{3}{2}} \]