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To solve the problem of finding the area between the curves \( p(x) \) and \( g(x) \), let's go through the necessary steps.

### 1. **Define the functions:**

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

### 2. **Find the points of intersection:**

To find the limits of integration, we need to solve the equation \( p(x) = g(x) \):

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

Simplify by subtracting \( \frac{2}{7}x + 1 \) from both sides:

\[
-\frac{4}{49}x^2 - \frac{2}{7}x + \frac{2}{7}x + 1 - 1 = 0
\]
\[
-\frac{4}{49}x^2 = 0
\]

This implies that \( x = 0 \). So, the curves intersect at \( x = 0 \).

### 3. **Set up the integral for the area:**

Since the curves intersect at \( x = 0 \), we would integrate over the interval where \( p(x) \) is below \( g(x) \). 

However, to find any other points of intersection (though from visual inspection it seems \( x = 0 \) might be the only one), we solve the quadratic. But in this case, \( x = 0 \) is indeed the only solution. Hence, you can consider the limits of integration from \( x = 0 \) to where the parabola \( p(x) \) touches the x-axis.

### 4. **Compute the area:**

To calculate the area between these two curves from \( x = 0 \) to the point where they intersect, we subtract the lower function \( p(x) \) from the upper function \( g(x) \):

\[
\text{Area} = \int_{0}^{b} (g(x) - p(x)) \, dx
\]

But as \( g(x) \) and \( p(x) \) are identical over \( x=0 \), the area might involve evaluating over any intercept of \( p(x) \) on the x-axis (which needs solving).

Since \( p(x) = g(x) \) at \( x = 0 \) but involves a parabola, the area involves solving the quadratic completely for intercepts.

However, given the direct setup:

The correct integral then gives:
\[ \int_{0}^{7} \left[\left(\frac{2}{7}x + 1\right) - \left(-\frac{4}{49}x^2 - \frac{2}{7}x + 1 \right)\right] dx \]

Let's compute this integral.

### 5. **Solve the integral:**
\[
A = \int_{0}^{7} \left(\frac{4}{49}x^2 + \frac{4}{7}x \right) dx 
= \left[\frac{4}{49}\frac{x^3}{3} + \frac{4}{7}\frac{x^2}{2} \right]_0^7 
\]

Evaluating this:

\[
A = \frac{4}{49} \times \frac{343}{3} + \frac{4}{7} \times \frac{49}{2}
= \frac{1372}{147} + \frac{196}{14}
= 9.33
\]

So, the area of the shaded region is approximately \(9.33 \, \text{units}^2 \).

Would you like any more detailed steps, or have any questions?

### Related Questions:
1. How would the solution change if \( p(x) \) was modified slightly?
2. What if \( g(x) \) was a quadratic function too?
3. How would you handle the area if the functions intersected at more than two points?
4. Could you solve this problem using numerical methods instead of integration?
5. How does the integral change if you rotate the curves about the y-axis?
6. What would be the impact on the area if \( p(x) \) had a positive coefficient for \( x^2 \)?
7. How can you verify the area calculated here using a different method?
8. How would the result differ if the parabola opened upwards?

### Tip:
Remember to double-check the points of intersection before setting up the limits of integration. This ensures accuracy in area calculations between curves.

Learn how to calculate the area between curves using integration with a step-by-step example. This problem involves finding the area between the curves of a parabola and a linear function. Suitable for advanced high school students (Grades 11-12) studying calculus.

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