The area bounded by the curve `y =4 sin x, ` x-axis from `x =0` to ` x = pi` is equal to :

Options

A. 1 sq. units
B. 2 sq. units
C. 4 sq. units
D. 8 sq units

Answer

To find the area bounded by the curve \( y = 4 \sin x \), the x-axis, from \( x = 0 \) to \( x = \pi \), we can follow these steps:

### Step 1: Set up the integral for the area
The area \( A \) under the curve from \( x = 0 \) to \( x = \pi \) can be calculated using the definite integral:
\[ A = \int_{0}^{\pi} 4 \sin x \, dx \]

### Step 2: Calculate the integral
Now, we will compute the integral:
\[ A = 4 \int_{0}^{\pi} \sin x \, dx \]

### Step 3: Find the antiderivative of \( \sin x \)
The antiderivative of \( \sin x \) is \( -\cos x \). Therefore, we can evaluate the integral:
\[ A = 4 \left[ -\cos x \right]_{0}^{\pi} \]

### Step 4: Evaluate the definite integral
Now, we will substitute the limits into the antiderivative:
\[ A = 4 \left[ -\cos(\pi) - (-\cos(0)) \right] \]
\[ = 4 \left[ -(-1) - (-1) \right] \]
\[ = 4 \left[ 1 - (-1) \right] \]
\[ = 4 \left[ 1 + 1 \right] \]
\[ = 4 \times 2 = 8 \]

### Final Answer
Thus, the area bounded by the curve \( y = 4 \sin x \), the x-axis, from \( x = 0 \) to \( x = \pi \) is:
\[ \boxed{8} \]

---