To solve this problem, let’s break it down into two parts:

1. Understanding the Function and Region Boundaries

Given:

• The function is $y = 7 \cos(x)$.
• The bounds are $x = -\frac{\pi}{2}$ and $x = \pi$.

To find the net area and the area of the region bounded by $y = 7 \cos(x)$ and the $x$-axis, we need to determine where the function crosses the $x$-axis within the given interval. This occurs where $\cos(x) = 0$, which happens at specific points within our interval.

2. Graphing the Function $y = 7 \cos(x)$

To visualize, let’s graph $y = 7 \cos(x)$ between $x = -\frac{\pi}{2}$ and $x = \pi$.

1. Identify Zero Crossings:

   • $\cos(x) = 0$ when $x = \frac{\pi}{2}$ within our interval.

2. Analyze Sign Changes:

   • From $x = -\frac{\pi}{2}$ to $x = \frac{\pi}{2}$, $y = 7 \cos(x)$ is positive.
   • From $x = \frac{\pi}{2}$ to $x = \pi$, $y = 7 \cos(x)$ is negative.

3. Setting Up the Integral for the Net Area

The net area between $x = -\frac{\pi}{2}$ and $x = \pi$ is given by the integral:

$\text{Net Area} = \int_{-\frac{\pi}{2}}^{\pi} 7 \cos(x) \, dx$

4. Setting Up the Integral for the Total Bounded Area

For the total area, we take the absolute value of $y = 7 \cos(x)$ across each interval:

$\text{Total Area} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 7 \cos(x) \, dx + \int_{\frac{\pi}{2}}^{\pi} |7 \cos(x)| \, dx$

Because $\cos(x)$ is negative in $\left(\frac{\pi}{2}, \pi\right)$, we can rewrite this as:

$\text{Total Area} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 7 \cos(x) \, dx - \int_{\frac{\pi}{2}}^{\pi} 7 \cos(x) \, dx$

5. Solving the Integrals

Let’s calculate each integral separately:

1. Net Area Integral:

   $\int_{-\frac{\pi}{2}}^{\pi} 7 \cos(x) \, dx = 7 \left[\sin(x)\right]_{-\frac{\pi}{2}}^{\pi} = 7 \left(\sin(\pi) - \sin\left(-\frac{\pi}{2}\right)\right) = 7 \left(0 + 1\right) = 7$

2. Total Area Integrals:

   For the positive part: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 7 \cos(x) \, dx = 7 \left[\sin(x)\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = 7 \left(1 - (-1)\right) = 7 \times 2 = 14$

   For the negative part: $\int_{\frac{\pi}{2}}^{\pi} -7 \cos(x) \, dx = 7 \left[-\sin(x)\right]_{\frac{\pi}{2}}^{\pi} = 7 \left(0 - 1\right) = 7$

So, the total area is:

$14 + 7 = 21$

Answer Summary

• Net Area = 7
• Total Bounded Area = 21

Would you like a graph of the function along with these results, or further clarification on any step?

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Related Questions

1. How do we apply this method to other trigonometric functions, like $y = \sin(x)$?
2. What is the difference between net area and total bounded area in integral calculus?
3. How does symmetry help in calculating areas for functions like $\cos(x)$ or $\sin(x)$?
4. How would the calculation change if the function was $y = 7 \sin(x)$ instead of $y = 7 \cos(x)$?
5. Can this method be used to calculate areas under non-periodic functions as well?

Tip:

When integrating absolute values of functions, breaking down the integration intervals based on sign changes can simplify calculations significantly.