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The problem asks us to find the area enclosed by the curves \( y = \sec^2{x} \), \( y = \tan^2{x} \), and the vertical lines \( x = -\frac{\pi}{4} \) and \( x = \frac{\pi}{4} \), as depicted in the image.

### Steps to solve the problem:

1. **Identify the curves and bounds:**
   - The region of interest is between \( x = -\frac{\pi}{4} \) and \( x = \frac{\pi}{4} \).
   - The upper curve is \( y = \sec^2{x} \).
   - The lower curve is \( y = \tan^2{x} \).

2. **Set up the integral:**
   To find the area between two curves, we compute the integral of the difference between the upper and lower functions over the given interval:
   \[
   \text{Area} = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (\sec^2{x} - \tan^2{x}) \, dx
   \]

3. **Simplify the integrand:**
   Using the identity \( \sec^2{x} - \tan^2{x} = 1 \), we can simplify the integral:
   \[
   \text{Area} = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 1 \, dx
   \]
   This is simply the integral of 1 over the interval \( \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \).

4. **Compute the integral:**
   \[
   \text{Area} = \left[ x \right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} = \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{2}
   \]

Thus, the area enclosed by the curves is \( \frac{\pi}{2} \).

Would you like further clarification on any step?

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Here are 5 related questions you might explore:
1. How do trigonometric identities help simplify integrals in calculus?
2. What are other common methods for finding the area between two curves?
3. Can we find areas involving more complex trigonometric functions?
4. How do you determine which curve is the upper or lower function in a given region?
5. How does the process change if the limits of integration are different?

**Tip:** Trigonometric identities are powerful tools in calculus, often simplifying complex integrals and allowing for more efficient problem-solving.

Find the areas of the regions enclosed by the lines and curves: y = sec^2(x), y = tan^2(x), x = -π/4, and x = π/4.

Learn how to compute the area enclosed by the curves y = sec^2(x) and y = tan^2(x) between x = -π/4 and x = π/4. This solution involves the use of trigonometric identities and definite integrals, suitable for calculus students.

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