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To find the area of the shaded region between the curves \( p(x) \) and \( g(x) \), we can use the following steps:

### Given Functions:
\[
p(x) = \frac{4}{49}x^2 - \frac{2}{7}x + 1
\]
\[
g(x) = \frac{2}{7}x + 1
\]

### Step 1: Find the Points of Intersection
We need to find the points where the two functions intersect by setting \( p(x) = g(x) \).

\[
\frac{4}{49}x^2 - \frac{2}{7}x + 1 = \frac{2}{7}x + 1
\]

Simplifying by subtracting \( g(x) \) from both sides:

\[
\frac{4}{49}x^2 - \frac{2}{7}x - \frac{2}{7}x = 0
\]

This simplifies to:

\[
\frac{4}{49}x^2 - \frac{4}{7}x = 0
\]

Factor out \( \frac{4}{49}x \):

\[
\frac{4}{49}x(x - 7) = 0
\]

So, the solutions are \( x = 0 \) and \( x = 7 \).

### Step 2: Set Up the Integral
The area \( A \) of the shaded region between \( x = 0 \) and \( x = 7 \) can be found using the integral:

\[
A = \int_{0}^{7} \left[ g(x) - p(x) \right] \, dx
\]

Substitute the given functions:

\[
A = \int_{0}^{7} \left[\left(\frac{2}{7}x + 1\right) - \left(\frac{4}{49}x^2 - \frac{2}{7}x + 1\right)\right] \, dx
\]

Simplifying the integrand:

\[
A = \int_{0}^{7} \left[ \frac{2}{7}x + 1 - \frac{4}{49}x^2 + \frac{2}{7}x - 1 \right] \, dx
\]

\[
A = \int_{0}^{7} \left[ \frac{4}{7}x - \frac{4}{49}x^2 \right] \, dx
\]

### Step 3: Compute the Integral
Separate the integral:

\[
A = \int_{0}^{7} \frac{4}{7}x \, dx - \int_{0}^{7} \frac{4}{49}x^2 \, dx
\]

Evaluate each integral:

\[
\int \frac{4}{7}x \, dx = \frac{4}{7} \cdot \frac{x^2}{2} = \frac{2}{7}x^2
\]

\[
\int \frac{4}{49}x^2 \, dx = \frac{4}{49} \cdot \frac{x^3}{3} = \frac{4}{147}x^3
\]

Now, plug in the limits from 0 to 7:

\[
A = \left[\frac{2}{7}x^2\right]_{0}^{7} - \left[\frac{4}{147}x^3\right]_{0}^{7}
\]

Evaluate:

\[
A = \left(\frac{2}{7} \cdot 7^2\right) - \left(\frac{4}{147} \cdot 7^3\right)
\]

\[
A = \left(\frac{2}{7} \cdot 49\right) - \left(\frac{4}{147} \cdot 343\right)
\]

\[
A = 14 - \frac{1372}{147}
\]

\[
A = 14 - \frac{392}{49} = 14 - 8 = 6 \text{ square units}
\]

So, the area of the shaded region is **6 square units**.

Would you like more details or have any questions?

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Here are some related questions:

1. What is the significance of finding the area between two curves?
2. How would the problem change if the functions intersected at more points?
3. Can you calculate the area under a curve using definite integrals?
4. How would you solve this problem if the region was unbounded?
5. What is the role of the limits of integration in this problem?
6. Could this area be found using numerical methods? If so, how?
7. How would the integral change if the curves were not functions but parametrically defined?
8. How do you interpret the area result geometrically?

**Tip:** When finding the area between two curves, ensure that you correctly identify which curve is on top and which is on the bottom over the interval of integration.

Learn how to calculate the area between curves using definite integrals. Step-by-step solution for finding the area of the shaded region between two given functions. Suitable for advanced high school students studying integration and calculus.

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