The area enclosed by the curves`y= sinx+cosx and y = | cosx-sin x |` over the interval `[0,pi/2]`

Options

A. `4(sqrt(2)-1)`
B. `2sqrt(2)(sqrt(2)-1)`
C. `2(sqrt(2)+1)`
D. `2sqrt(2)(sqrt(2)+1)`

Explanation

To find the area enclosed by the curves \( y = \sin x + \cos x \) and \( y = |\cos x - \sin x| \) over the interval \([0, \frac{\pi}{2}]\), we will follow these steps:

### Step 1: Identify the curves and their intersection points
The curves are:
1. \( y_1 = \sin x + \cos x \)
2. \( y_2 = |\cos x - \sin x| \)

We need to find the points where these two curves intersect within the interval \([0, \frac{\pi}{2}]\).

### Step 2: Determine the behavior of \( |\cos x - \sin x| \)
In the interval \([0, \frac{\pi}{2}]\):
- At \( x = 0 \): \( \cos(0) - \sin(0) = 1 \) (positive)
- At \( x = \frac{\pi}{4} \): \( \cos(\frac{\pi}{4}) - \sin(\frac{\pi}{4}) = 0 \) (zero)
- At \( x = \frac{\pi}{2} \): \( \cos(\frac{\pi}{2}) - \sin(\frac{\pi}{2}) = -1 \) (negative)

Thus, \( |\cos x - \sin x| = \cos x - \sin x \) for \( x \in [0, \frac{\pi}{4}] \) and \( |\cos x - \sin x| = \sin x - \cos x \) for \( x \in [\frac{\pi}{4}, \frac{\pi}{2}] \).

### Step 3: Set up the integral for the area
The area \( A \) can be expressed as:
\[ A = \int_0^{\frac{\pi}{4}} \left( (\sin x + \cos x) - (\cos x - \sin x) \right) dx + \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \left( (\sin x + \cos x) - (\sin x - \cos x) \right) dx \]

### Step 4: Simplify the integrals
1. For \( x \in [0, \frac{\pi}{4}] \):
\[ A_1 = \int_0^{\frac{\pi}{4}} (2\sin x) \, dx \]

2. For \( x \in [\frac{\pi}{4}, \frac{\pi}{2}] \):
\[ A_2 = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (2\cos x) \, dx \]

### Step 5: Calculate the integrals
1. Calculate \( A_1 \):
\[ A_1 = 2 \int_0^{\frac{\pi}{4}} \sin x \, dx = 2 \left[-\cos x\right]_0^{\frac{\pi}{4}} = 2 \left(-\frac{1}{\sqrt{2}} + 1\right) = 2 \left(1 - \frac{1}{\sqrt{2}}\right) \]

2. Calculate \( A_2 \):
\[ A_2 = 2 \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cos x \, dx = 2 \left[\sin x\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} = 2 \left(1 - \frac{1}{\sqrt{2}}\right) \]

### Step 6: Combine the areas
\[ A = A_1 + A_2 = 2 \left(1 - \frac{1}{\sqrt{2}}\right) + 2 \left(1 - \frac{1}{\sqrt{2}}\right) = 4 \left(1 - \frac{1}{\sqrt{2}}\right) \]

### Step 7: Final area expression
Thus, the total area enclosed by the curves is:
\[ A = 4 \left(1 - \frac{1}{\sqrt{2}}\right) \]