Find the area of the region bounded by the curve ` y = sin x ` between ` x = 0 and x= 2pi `.

Answer

To find the area of the region bounded by the curve \( y = \sin x \) between \( x = 0 \) and \( x = 2\pi \), we can follow these steps:

### Step 1: Understand the Curve
The function \( y = \sin x \) oscillates between -1 and 1. Between \( x = 0 \) and \( x = \pi \), the sine function is positive, and between \( x = \pi \) and \( x = 2\pi \), it is negative.

### Step 2: Set Up the Integral
To find the area, we need to calculate the integral of \( \sin x \) from \( 0 \) to \( 2\pi \). However, since the area below the x-axis (from \( \pi \) to \( 2\pi \)) will contribute negatively, we will need to take the absolute value of the integral from \( \pi \) to \( 2\pi \).

### Step 3: Calculate the Area from \( 0 \) to \( \pi \)
First, we calculate the area from \( 0 \) to \( \pi \):
\[ \text{Area}_1 = \int_0^{\pi} \sin x \, dx \]

### Step 4: Compute the Integral
The integral of \( \sin x \) is:
\[ \int \sin x \, dx = -\cos x \]
Now, we evaluate it from \( 0 \) to \( \pi \):
\[ \text{Area}_1 = \left[-\cos x\right]_0^{\pi} = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2 \]

### Step 5: Calculate the Area from \( \pi \) to \( 2\pi \)
Next, we calculate the area from \( \pi \) to \( 2\pi \):
\[ \text{Area}_2 = \int_{\pi}^{2\pi} \sin x \, dx \]

### Step 6: Compute the Integral
Using the same integral:
\[ \text{Area}_2 = \left[-\cos x\right]_{\pi}^{2\pi} = -\cos(2\pi) - (-\cos(\pi)) = -1 - 1 = -2 \]
Since this area is below the x-axis, we take the absolute value:
\[ \text{Area}_2 = 2 \]

### Step 7: Total Area
Now, we add both areas to find the total area:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = 2 + 2 = 4 \]

### Final Answer
The area of the region bounded by the curve \( y = \sin x \) between \( x = 0 \) and \( x = 2\pi \) is \( 4 \) square units.

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