Question

The area (in sq. units) of the region bounded by the curves \( y = 4 \cos x \) and \( y = -|\cos x| \) from \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \) is:

• A. 6
• B. 8
• C. 12
• D. 10

Hint:

When dealing with absolute value functions, break them into pieces based on their sign, and then compute the area by integrating the difference of the functions.

Answer 1

The Correct Option is D

We are given two curves: \[ y = 4 \cos x \quad \text{and} \quad y = -|\cos x| \] The area enclosed between these curves can be found by integrating the difference of the two curves over the interval \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \). First, observe that \( y = -|\cos x| \) will be equivalent to \( y = -\cos x \) in the interval \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \), since cosine is non-negative in the given range. The area is given by: \[ \text{Area} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left[ 4 \cos x - (-\cos x) \right] dx \] \[ \text{Area} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 5 \cos x \, dx \] Now, integrate: \[ \text{Area} = 5 \left[ \sin x \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \] \[ \text{Area} = 5 \left[ \sin\left(\frac{\pi}{2}\right) - \sin\left(-\frac{\pi}{2}\right) \right] \] \[ \text{Area} = 5 \left[ 1 - (-1) \right] = 5 \times 2 = 10 \] Thus, the correct answer is option (4), \( \text{Area} = 10 \).

Answer 2

Step 1: Understand the curves and the interval
We have two curves:
\( y = 4 \cos x \) and \( y = -|\cos x| \),
and the interval for \(x\) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 2: Analyze the behavior of the curves
- \( \cos x \) is positive in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), so \( |\cos x| = \cos x \) in this interval.
- Thus, \( y = -|\cos x| = -\cos x \) for \( x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

Step 3: Identify the area between the curves
The region bounded by the curves is between:
Upper curve: \( y = 4 \cos x \)
Lower curve: \( y = - \cos x \)

Step 4: Set up the integral for the area
Area \(A\) = \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left[ (4 \cos x) - (- \cos x) \right] dx = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 5 \cos x \, dx\).

Step 5: Calculate the integral
\[ A = 5 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx = 5 \left[ \sin x \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = 5 (\sin \frac{\pi}{2} - \sin -\frac{\pi}{2}) = 5 (1 - (-1)) = 5 \times 2 = 10 \].

Final Answer:
The area of the region bounded by the curves is \(10\) square units.