The area bounded by `y=cos x and x=-pi/2 and x=2pi`.and the axis of xin square units is

Options

A. 4
B. 5
C. 6
D. 7

Answer

To find the area bounded by the curve \( y = \cos x \), the lines \( x = -\frac{\pi}{2} \) and \( x = 2\pi \), and the x-axis, we will follow these steps:

### Step 1: Set up the integral for the area
The area \( A \) under the curve from \( x = a \) to \( x = b \) can be calculated using the integral:
\[ A = \int_{a}^{b} y \, dx \]
In this case, \( y = \cos x \), \( a = -\frac{\pi}{2} \), and \( b = 2\pi \).

### Step 2: Identify the intervals of integration
The function \( \cos x \) oscillates between positive and negative values. We need to determine where \( \cos x \) is positive and negative within the interval \( [-\frac{\pi}{2}, 2\pi] \).

1. From \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \), \( \cos x \) is positive.
2. From \( x = \frac{\pi}{2} \) to \( x = \frac{3\pi}{2} \), \( \cos x \) is negative.
3. From \( x = \frac{3\pi}{2} \) to \( x = 2\pi \), \( \cos x \) is positive again.

### Step 3: Break the integral into parts
We can break the integral into three parts based on the intervals identified:
\[ A = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx + \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos x \, dx + \int_{\frac{3\pi}{2}}^{2\pi} \cos x \, dx \]

### Step 4: Calculate each integral
1. **First Integral**:
\[ A_1 = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx \]
\[ = [\sin x]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin\left(-\frac{\pi}{2}\right) = 1 - (-1) = 2 \]

2. **Second Integral**:
\[ A_2 = \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos x \, dx \]
\[ = [\sin x]_{\frac{\pi}{2}}^{\frac{3\pi}{2}} = \sin\left(\frac{3\pi}{2}\right) - \sin\left(\frac{\pi}{2}\right) = -1 - 1 = -2 \]
Since area cannot be negative, we take the absolute value:
\[ A_2 = 2 \]

3. **Third Integral**:
\[ A_3 = \int_{\frac{3\pi}{2}}^{2\pi} \cos x \, dx \]
\[ = [\sin x]_{\frac{3\pi}{2}}^{2\pi} = \sin(2\pi) - \sin\left(\frac{3\pi}{2}\right) = 0 - (-1) = 1 \]

### Step 5: Sum the areas
Now, we can sum the areas from each part:
\[ A = A_1 + |A_2| + A_3 = 2 + 2 + 1 = 5 \]

### Final Answer
The area bounded by the curve \( y = \cos x \), the lines \( x = -\frac{\pi}{2} \) and \( x = 2\pi \), and the x-axis is:
\[ \boxed{5} \text{ square units} \]