What is the area of the one of the loops between the curve y=c sin x and x-axis?

Options

A. c
B. 2c
C. 3c
D. 4c

Answer

To find the area of one of the loops between the curve \( y = c \sin x \) and the x-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Loop**:
The function \( y = c \sin x \) oscillates between \( -c \) and \( c \). The loops occur between the x-axis and the curve, specifically from \( x = 0 \) to \( x = \pi \) for one complete loop.

2. **Set Up the Area Integral**:
The area \( A \) of one loop can be calculated using the definite integral:
\[ A = \int_{0}^{\pi} c \sin x \, dx \]

3. **Factor Out the Constant**:
Since \( c \) is a constant, we can factor it out of the integral:
\[ A = c \int_{0}^{\pi} \sin x \, dx \]

4. **Integrate \( \sin x \)**:
The integral of \( \sin x \) is \( -\cos x \). Therefore:
\[ A = c \left[ -\cos x \right]_{0}^{\pi} \]

5. **Evaluate the Integral**:
Now, we will evaluate the definite integral from \( 0 \) to \( \pi \):
\[ A = c \left( -\cos(\pi) - (-\cos(0)) \right) \]
Knowing that \( \cos(\pi) = -1 \) and \( \cos(0) = 1 \):
\[ A = c \left( -(-1) - (-1) \right) = c \left( 1 + 1 \right) = c \cdot 2 \]

6. **Final Result**:
Thus, the area of one of the loops is:
\[ A = 2c \]

### Final Answer:
The area of one of the loops between the curve \( y = c \sin x \) and the x-axis is \( 2c \).

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