The area bounded by `y=x+1 and y=cos x` and the x - axis, is

Options

A. 1 sq. units
B. `(3)/(2)" sq. units"`
C. `(1)/(4)" sq. units"`
D. `(1)/(8)" sq. units"`

Answer

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

### Step 1: Identify the points of intersection
We need to find the points where the curves \( y = x + 1 \) and \( y = \cos x \) intersect. To do this, we set them equal to each other:

\[ x + 1 = \cos x \]

This equation cannot be solved algebraically, so we will find the points of intersection graphically or numerically.

### Step 2: Graph the functions
We can sketch the graphs of \( y = x + 1 \) and \( y = \cos x \) to identify the points of intersection. The graph of \( y = \cos x \) oscillates between -1 and 1, while the line \( y = x + 1 \) is a straight line with a slope of 1.

From the graph, we can observe that the curves intersect at approximately \( x = -1 \) and \( x = 0 \).

### Step 3: Set up the integral for the area
The area \( A \) between the curves from \( x = -1 \) to \( x = 0 \) can be calculated using the integral:

\[ A = \int_{-1}^{0} (\cos x - (x + 1)) \, dx \]

### Step 4: Simplify the integrand
We simplify the integrand:

\[ \cos x - (x + 1) = \cos x - x - 1 \]

### Step 5: Evaluate the integral
Now we will compute the integral:

\[ A = \int_{-1}^{0} (\cos x - x - 1) \, dx \]

This can be split into three separate integrals:

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

Calculating each integral separately:

1. **Integral of \( \cos x \)**:
\[ \int \cos x \, dx = \sin x \quad \text{so} \quad \left[ \sin x \right]_{-1}^{0} = \sin(0) - \sin(-1) = 0 - (-\sin(1)) = \sin(1) \]

2. **Integral of \( x \)**:
\[ \int x \, dx = \frac{x^2}{2} \quad \text{so} \quad \left[ \frac{x^2}{2} \right]_{-1}^{0} = 0 - \frac{(-1)^2}{2} = -\frac{1}{2} \]

3. **Integral of 1**:
\[ \int 1 \, dx = x \quad \text{so} \quad \left[ x \right]_{-1}^{0} = 0 - (-1) = 1 \]

Putting it all together:

\[ A = \sin(1) - \left(-\frac{1}{2}\right) - 1 \]
\[ A = \sin(1) + \frac{1}{2} - 1 = \sin(1) - \frac{1}{2} \]

### Final Step: Calculate the area
Thus, the area bounded by the curves and the x-axis is:

\[ A = \sin(1) - \frac{1}{2} \]