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To solve the problem of finding the area bounded above by \( p(x) \) and below by \( q(x) \), we need to set up the integral of the difference between these two functions over the interval where they intersect.

### Step 1: Determine the Intersection Points
First, we need to find the points where \( p(x) \) and \( q(x) \) intersect. These points will serve as the limits of integration.

Given:
\[ p(x) = -4\sqrt{\frac{-x + 1}{7}} + 2 \]
\[ q(x) = -\frac{4x + 18}{7} \]

Set \( p(x) \) equal to \( q(x) \) and solve for \( x \):
\[ -4\sqrt{\frac{-x + 1}{7}} + 2 = -\frac{4x + 18}{7} \]

Multiply both sides by 7 to clear the fraction:
\[ -28\sqrt{\frac{-x + 1}{7}} + 14 = -4x - 18 \]

Rearrange to isolate the square root term:
\[ -28\sqrt{\frac{-x + 1}{7}} = -4x - 32 \]
\[ 28\sqrt{\frac{-x + 1}{7}} = 4x + 32 \]

Divide by 4:
\[ 7\sqrt{\frac{-x + 1}{7}} = x + 8 \]

Square both sides to eliminate the square root:
\[ 49 \cdot \frac{-x + 1}{7} = (x + 8)^2 \]
\[ 7(-x + 1) = x^2 + 16x + 64 \]
\[ -7x + 7 = x^2 + 16x + 64 \]

Rearrange to form a quadratic equation:
\[ x^2 + 23x + 57 = 0 \]

Solve the quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-23 \pm \sqrt{23^2 - 4 \cdot 1 \cdot 57}}{2 \cdot 1} \]
\[ x = \frac{-23 \pm \sqrt{529 - 228}}{2} \]
\[ x = \frac{-23 \pm \sqrt{301}}{2} \]

So the intersection points are:
\[ x_1 = \frac{-23 + \sqrt{301}}{2} \]
\[ x_2 = \frac{-23 - \sqrt{301}}{2} \]

### Step 2: Set Up the Integral
The area \( A \) between \( p(x) \) and \( q(x) \) is given by:
\[ A = \int_{x_1}^{x_2} [q(x) - p(x)] \, dx \]

### Step 3: Calculate the Integral
Evaluate the integral from \( x_1 \) to \( x_2 \):
\[ \int_{x_1}^{x_2} \left( -\frac{4x + 18}{7} + 4\sqrt{\frac{-x + 1}{7}} - 2 \right) dx \]

Let's compute this integral step-by-step.

1. Find the antiderivative of each term:
   - For \( -\frac{4x + 18}{7} \), use basic polynomial integration.
   - For \( 4\sqrt{\frac{-x + 1}{7}} \), use a substitution method.
   - For \( -2 \), use basic polynomial integration.

2. Evaluate the antiderivatives at the limits \( x_1 \) and \( x_2 \).

### Tip: Check your algebraic manipulations carefully and ensure that all steps follow logically from the previous ones.

If you need more detailed steps for each part of the integral or have any other questions, let me know! Here are some related questions you might find helpful:

1. How do you find the area between two curves?
2. What is the significance of intersection points in determining the bounded area?
3. How does changing the order of integration affect the result?
4. What are the common methods for solving integrals involving square roots?
5. How do you verify the solution of a definite integral?
6. What is the difference between definite and indefinite integrals?
7. How can you visualize the area under a curve to understand the problem better?
8. What are the applications of finding areas between curves in real-world problems?

Learn how to calculate the area between two curves using the integral method. Step-by-step solution to finding the area bounded by p(x) and q(x), including finding intersection points and evaluating integrals. Suitable for advanced high school students and calculus learners.

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