Area of the region bounded by `y=|5sinx|` from x=0 to `x=4pi` and x-axis is

Options

A. 10
B. 20
C. 40
D. 80

Answer

To find the area of the region bounded by the curve \( y = |5 \sin x| \) from \( x = 0 \) to \( x = 4\pi \) and the x-axis, we will follow these steps:

### Step 1: Set up the integral for the area
The area \( A \) can be calculated using the integral:
\[ A = \int_0^{4\pi} |5 \sin x| \, dx \]

### Step 2: Analyze the function \( |5 \sin x| \)
The function \( \sin x \) oscillates between -1 and 1. Therefore, \( |5 \sin x| \) will oscillate between 0 and 5. The sine function completes one full cycle from \( 0 \) to \( 2\pi \). Since we are integrating from \( 0 \) to \( 4\pi \), we will have two complete cycles of the sine function.

### Step 3: Break the integral into intervals
We can break the integral into two parts, each covering one complete cycle:
\[ A = \int_0^{2\pi} |5 \sin x| \, dx + \int_{2\pi}^{4\pi} |5 \sin x| \, dx \]
Since \( |5 \sin x| \) has the same shape in both intervals, we can simplify this to:
\[ A = 2 \int_0^{2\pi} |5 \sin x| \, dx \]

### Step 4: Evaluate the integral from \( 0 \) to \( 2\pi \)
Within the interval \( [0, 2\pi] \), \( \sin x \) is positive from \( 0 \) to \( \pi \) and negative from \( \pi \) to \( 2\pi \). Thus, we can write:
\[ A = 2 \left( \int_0^{\pi} 5 \sin x \, dx + \int_{\pi}^{2\pi} -5 \sin x \, dx \right) \]

### Step 5: Calculate the integrals
Calculating the first integral:
\[ \int_0^{\pi} 5 \sin x \, dx = 5 \left[ -\cos x \right]_0^{\pi} = 5 \left( -\cos \pi + \cos 0 \right) = 5 \left( 1 + 1 \right) = 10 \]

Calculating the second integral:
\[ \int_{\pi}^{2\pi} -5 \sin x \, dx = -5 \left[ -\cos x \right]_{\pi}^{2\pi} = -5 \left( -\cos(2\pi) + \cos(\pi) \right) = -5 \left( 1 - (-1) \right) = -5 \times 2 = -10 \]

### Step 6: Combine the results
Now, substituting back into the area formula:
\[ A = 2 \left( 10 + 10 \right) = 2 \times 20 = 40 \]

### Final Answer
Thus, the area of the region bounded by \( y = |5 \sin x| \) from \( x = 0 \) to \( x = 4\pi \) and the x-axis is:
\[ \boxed{40} \text{ square units} \]