The area of the loop between the curve `y=asinx` and x-axis is (A) `a` (B) `2a` (C) `3a` (D) none of these

Options

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

Answer

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

### Step 1: Understand the Curve
The curve \( y = a \sin x \) oscillates between \( -a \) and \( a \). The sine function has a period of \( 2\pi \), but for one loop (from the x-axis to the peak and back to the x-axis), we only need to consider the interval from \( 0 \) to \( \pi \).

### Step 2: Set Up the Integral
The area \( A \) between the curve and the x-axis from \( x = 0 \) to \( x = \pi \) can be calculated using the definite integral:
\[ A = \int_{0}^{\pi} y \, dx = \int_{0}^{\pi} a \sin x \, dx \]

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

### Step 4: Integrate \( \sin x \)
The integral of \( \sin x \) is:
\[ \int \sin x \, dx = -\cos x \]
Thus, we evaluate:
\[ \int_{0}^{\pi} \sin x \, dx = \left[-\cos x\right]_{0}^{\pi} \]

### Step 5: Evaluate the Integral
Now we substitute the limits into the evaluated integral:
\[ = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2 \]

### Step 6: Calculate the Area
Substituting back into our area formula:
\[ A = a \cdot 2 = 2a \]

### Conclusion
The area of the loop between the curve \( y = a \sin x \) and the x-axis is \( 2a \).

### Final Answer
The correct option is (B) \( 2a \).

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