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To solve the problem of finding the area between the curves \( p(x) \) and \( g(x) \), follow these steps:

### 1. **Define the functions:**
The functions provided are:
\[
p(x) = -\frac{4}{49}x^2 - \frac{2}{7}x + 1
\]
\[
g(x) = \frac{2}{7}x + 1
\]

### 2. **Find the points of intersection:**
The area of interest is between the points where the two curves intersect. We find these points by setting \( p(x) \) equal to \( g(x) \):

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

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 \( x = 0 \), but this doesn't consider the fact that they intersect in another location given the quadratic. Let's set the expression fully and solve for intercepts.

For the intersection points:
\[
-\frac{4}{49}x^2 - \frac{2}{7}x + 1 = \frac{2}{7}x + 1
\]

Moving all terms to one side:
\[
-\frac{4}{49}x^2 - \frac{4}{7}x = 0
\]

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

This implies that \( x = 0 \) or \( x = -7 \).

### 3. **Set up the integral:**
The area \( A \) between the curves is given by the integral from \( x = -7 \) to \( x = 0 \):

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

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

Simplify the integrand:
\[
A = \int_{-7}^{0} \left(\frac{4}{49}x^2 + \frac{4}{7}x \right) dx
\]

### 4. **Solve the integral:**
Now, let's compute the integral:
\[
A = \int_{-7}^{0} \frac{4}{49}x^2 \, dx + \int_{-7}^{0} \frac{4}{7}x \, dx
\]

Solve each part:
- The first integral:
\[
\int \frac{4}{49}x^2 \, dx = \frac{4}{49} \cdot \frac{x^3}{3} = \frac{4}{147}x^3
\]
Evaluate from \( x = -7 \) to \( x = 0 \):
\[
\frac{4}{147}(0^3 - (-7)^3) = \frac{4}{147}(-343) = -\frac{1372}{147} = -9.33
\]

- The second integral:
\[
\int \frac{4}{7}x \, dx = \frac{4}{7} \cdot \frac{x^2}{2} = \frac{2}{7}x^2
\]
Evaluate from \( x = -7 \) to \( x = 0 \):
\[
\frac{2}{7}(0^2 - (-7)^2) = \frac{2}{7}(-49) = -14
\]

Add the two parts together:
\[
A = -9.33 + (-14) = -23.33 \text{ square units}
\]

Since area cannot be negative, we take the absolute value:
\[
A = 23.33 \text{ square units}
\]

### Conclusion:
The area of the shaded region is \( 23.33 \) square units.

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

### Related Questions:
1. How do you verify the area calculation by comparing with geometric methods?
2. Can this problem be solved using numerical integration instead?
3. How does the area calculation change if the curves intersect at multiple points?
4. What if the parabolic function was concave up, how would the solution differ?
5. How would you solve for the area if the functions were in terms of \( y \) instead of \( x \)?
6. How would this problem change if it was rotated about the y-axis?
7. What impact does the orientation of the parabola have on the area?
8. Could you approximate the area using a Riemann sum instead?

### Tip:
Always check for multiple points of intersection between curves before setting up integrals, especially when dealing with quadratic and linear functions.

Learn how to calculate the area between curves using integration. Step-by-step solution for finding the area between the curves represented by quadratic and linear functions. Includes detailed calculations and explanations.

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