Find the area of the region enclosed by the three graphs y = x2, y = 8 x, and y = 1

To find the area of the region enclosed by the three graphs \( y = x^2 \), \( y = 8x \), and \( y = 1 \), we need to follow these steps:

1. **Identify the points of intersection** of the curves to determine the boundaries of the region.
2. **Set up the integrals** to calculate the area between the curves.

### Step 1: Find the Points of Intersection
We need to find the points where the graphs intersect each other.

1. **Intersection of \( y = x^2 \) and \( y = 8x \):**
   \[
   x^2 = 8x \implies x^2 - 8x = 0 \implies x(x - 8) = 0
   \]
   So, the points of intersection are \( x = 0 \) and \( x = 8 \).

2. **Intersection of \( y = x^2 \) and \( y = 1 \):**
   \[
   x^2 = 1 \implies x = \pm 1
   \]
   So, the points of intersection are \( x = 1 \) and \( x = -1 \).

3. **Intersection of \( y = 8x \) and \( y = 1 \):**
   \[
   8x = 1 \implies x = \frac{1}{8}
   \]
   So, the point of intersection is \( x = \frac{1}{8} \).

### Step 2: Set Up the Integral for the Area
Now that we have the points of intersection, we can break the area calculation into different regions based on these points.

- **Region 1:** From \( x = -1 \) to \( x = 1 \), the area is between \( y = 1 \) and \( y = x^2 \).
- **Region 2:** From \( x = \frac{1}{8} \) to \( x = 8 \), the area is between \( y = 8x \) and \( y = x^2 \).

The total area is the sum of these two regions.

Let's compute the integrals for each region to find the enclosed area.The area of the region enclosed by the three graphs \( y = x^2 \), \( y = 8x \), and \( y = 1 \) is approximately **86.605 square units**.

Would you like more details or have any questions?

Here are some related questions to explore:
1. How would the area change if the equation \( y = 1 \) is replaced with \( y = 2 \)?
2. Can you find the area enclosed by \( y = x^2 \) and \( y = 8x \) alone, without \( y = 1 \)?
3. How would you find the centroid of the region bounded by these curves?
4. What happens to the area if \( y = 8x \) is replaced with \( y = 6x \)?
5. How do you determine the volume of the solid formed by rotating the enclosed area around the x-axis?

**Tip:** When finding areas between curves, always ensure the functions are correctly arranged to avoid negative areas.

Learn how to calculate the area enclosed by the curves y = x^2, y = 8x, and y = 1 using definite integrals. This problem involves finding points of intersection, setting up integrals, and applying the Fundamental Theorem of Calculus. Suitable for advanced high school students or calculus learners.

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