Find the net area and the area of the region bounded by \( y=7 \cos x \) and the \( x \)-axis between \( x=-\frac{\pi}{2} \) and \( x=\pi \). Graph the function and find the region indicated in this nuestinn

To find the area of the region bounded by \( y = 7 \cos x \) and the \( x \)-axis between \( x = -\frac{\pi}{2} \) and \( x = \pi \), we need to compute the definite integral of the function \( 7 \cos x \) over the interval, taking into account where the function is above and below the x-axis. 1. **Identifying Intervals**: - \( y = 7 \cos x \) is positive on the interval \( x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) and negative on \( x \in \left(\frac{\pi}{2}, \pi\right]\). 2. **Calculating Areas**: - For the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), the area is: \[ A_1 = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 7 \cos x \, dx \] - For the interval \( \left(\frac{\pi}{2}, \pi\right] \) where the graph is below the x-axis: \[ A_2 = \int_{\frac{\pi}{2}}^{\pi} -7 \cos x \, dx \] 3. **Evaluating the Integrals**: - **For \( A_1 \)**: \[ A_1 = 7 \left[ \sin x \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = 7 \left(\sin(\frac{\pi}{2}) - \sin(-\frac{\pi}{2})\right) = 7(1 - (-1)) = 7 \cdot 2 = 14 \] - **For \( A_2 \)**: \[ A_2 = 7 \left[ -\sin x \right]_{\frac{\pi}{2}}^{\pi} = 7 \left(-\sin(\pi) - (-\sin(\frac{\pi}{2}))\right) = 7(0 - (-1)) = 7 \cdot 1 = 7 \] 4. **Net Area**: \[ \text{Net Area} = A_1 + A_2 = 14 + 7 = 21 \] Therefore, the net area and the area of the region bounded by \( y = 7 \cos x \) and the \( x \)-axis between \( x = -\frac{\pi}{2} \) and \( x = \pi \) is 21 square units. Now, let's consider additional aspects: When \( y = 7 \cos x \) was first introduced in mathematics, it showcased the beauty of periodic functions and their applications in modeling oscillatory phenomena such as sound waves and light waves. The connection between trigonometric functions and physics made this exploration not only a mathematical endeavor but also a key to understanding the world around us. In the realm of practical applications, this type of integral helps engineers and scientists estimate areas under curves, a critical task in fields such as physics, architecture, and environmental science. For instance, understanding the profile of waveforms can influence design decisions in structures or materials that encounter periodic forces, like bridges or musical instruments. Integrals provide insights that curve not just through math but through real-world structures and systems!