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This task involves calculating the areas of the shaded regions for each of the given graphs using definite integrals of the given functions.

Here’s how we can approach this step-by-step:

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### **a) Calculation**

For the given functions:
- \( f(x) = 2^x \)
- \( g(x) = x^4 + 1 \)

The shaded region is bounded by the two functions between the given points of intersection. The area is given by:

\[
A = \int_{x_1}^{x_2} [f(x) - g(x)] \, dx
\]

Here, \( x_1 \) and \( x_2 \) are the points of intersection of \( f(x) \) and \( g(x) \). These can be determined by solving \( f(x) = g(x) \).

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### **b) Calculation**

For the given functions:
- \( f(x) = 0.25^x \)
- \( g(x) = -1.5x + 1.25 \)

The shaded region’s area is:

\[
A = \int_{x_1}^{x_2} [g(x) - f(x)] \, dx
\]

Again, solve \( f(x) = g(x) \) to find \( x_1 \) and \( x_2 \).

---

### **c) Calculation**

For the given functions:
- \( f(x) = 2 \cdot 0.5^x + x \)
- \( g(x) = -0.5x + 2.5 \)

The area of the shaded region is:

\[
A = \int_{x_1}^{x_2} [f(x) - g(x)] \, dx
\]

Here as well, solve \( f(x) = g(x) \) to find the points of intersection \( x_1 \) and \( x_2 \).

---

### Next Steps:
Would you like me to:
1. Solve one or more of these integrals step-by-step?
2. Determine the points of intersection for any of the graphs?
3. Use numerical methods to approximate the areas?
4. Explain integration techniques for these types of problems?
5. Provide a general explanation of definite integrals?

**Tip:** When solving such problems, always visualize which function is on top to set up the integral correctly.

Berechnen Sie den Inhalt der markierten Fläche mithilfe von Stammfunktionen.

Learn how to calculate the area between curves using definite integrals. This example involves exponential, polynomial, and linear functions, covering concepts such as integration, points of intersection, and definite integrals. Suitable for students in Grades 11-12.

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