The area between the curves `y=cosx,` x-axis and the line `y=x+1,` is

Options

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

Answer

To find the area between the curves \( y = \cos x \), the x-axis, and the line \( y = x + 1 \), we can follow these steps:

### Step 1: Identify the curves and points of intersection
We have two curves:
1. \( y = \cos x \)
2. \( y = x + 1 \)

To find the area between these curves, we first need to determine their points of intersection. We set the equations equal to each other:

\[ \cos x = x + 1 \]

### Step 2: Solve for points of intersection
To find the points of intersection, we can analyze the functions graphically or numerically. We can check for values of \( x \):

- At \( x = -1 \):
\[ \cos(-1) \approx 0.5403 \quad \text{and} \quad -1 + 1 = 0 \]
- At \( x = 0 \):
\[ \cos(0) = 1 \quad \text{and} \quad 0 + 1 = 1 \]

The curves intersect at \( x = -1 \) 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:

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

### Step 4: Simplify the integrand
The integrand simplifies to:

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

### Step 5: Calculate the integral
Now we compute the integral:

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

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:

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

2. \( \int x \, dx = \frac{x^2}{2} \)
\[ \int_{-1}^{0} x \, dx = \left[ \frac{x^2}{2} \right]_{-1}^{0} = 0 - \frac{(-1)^2}{2} = -\frac{1}{2} \]

3. \( \int 1 \, dx = x \)
\[ \int_{-1}^{0} 1 \, dx = [x]_{-1}^{0} = 0 - (-1) = 1 \]

### Step 6: Combine the results
Now we combine the results:

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

### Step 7: Final area calculation
Thus, the area is:

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

### Step 8: Numerical approximation
Using a calculator, we can find \( \sin(1) \approx 0.8415 \):

\[ A \approx 0.8415 - 0.5 = 0.3415 \]

### Conclusion
The area between the curves \( y = \cos x \), the x-axis, and the line \( y = x + 1 \) is approximately \( 0.3415 \) square units.