Find the area of the shaded region shown in the figure​ (which is not shown to​ scale).
x
y
y equals 52 minus x squared
y equals 9 x
y equals three sevenths x


To find the area of the shaded region between the curves, we need to:

1. Identify the boundaries (points of intersection) between the curves.
2. Set up the integrals for the area between the curves.
3. Calculate the area by integrating.

### 1. Identify Points of Intersection

The equations of the curves are:
- \( y = 52 - x^2 \)
- \( y = 9x \)
- \( y = \frac{3}{7}x \)

To find the points of intersection, solve for \( x \) by setting the equations equal to each other:

**Intersection of \( y = 52 - x^2 \) and \( y = 9x \):**
\[
52 - x^2 = 9x \quad \text{(rearranging to solve for } x \text{)}
\]
\[
x^2 + 9x - 52 = 0
\]

Solve the quadratic equation using the quadratic formula:
\[
x = \frac{-9 \pm \sqrt{9^2 - 4(1)(-52)}}{2(1)}
\]
\[
x = \frac{-9 \pm \sqrt{81 + 208}}{2}
\]
\[
x = \frac{-9 \pm \sqrt{289}}{2}
\]
\[
x = \frac{-9 \pm 17}{2}
\]
\[
x = \frac{-9 + 17}{2} = 4 \quad \text{or} \quad x = \frac{-9 - 17}{2} = -13
\]

Thus, the points of intersection are \( x = 4 \) and \( x = -13 \).

**Intersection of \( y = 9x \) and \( y = \frac{3}{7}x \):**
\[
9x = \frac{3}{7}x
\]
\[
\left( 9 - \frac{3}{7} \right) x = 0
\]
\[
\frac{60}{7}x = 0
\]

This gives \( x = 0 \), meaning the lines intersect at \( x = 0 \).

### 2. Set Up the Integral

Now that we have the points of intersection, we can set up the integral for the area. The area is computed as the integral of the upper curve minus the lower curve over the interval.

For the interval \( x = -13 \) to \( x = 4 \), the parabola \( y = 52 - x^2 \) is the upper curve, and \( y = 9x \) is the lower curve:
\[
\text{Area} = \int_{-13}^{4} \left( (52 - x^2) - 9x \right) dx
\]

For the region where \( x = 0 \) to \( x = 4 \), we use the two straight lines \( y = 9x \) and \( y = \frac{3}{7}x \):
\[
\text{Area} = \int_0^4 \left( 9x - \frac{3}{7}x \right) dx
\]

### 3. Compute the Integrals

Let's compute both integrals to find the total area:

\[
\text{Total Area} = \int_{-13}^{4} \left( (52 - x^2) - 9x \right) dx + \int_0^4 \left( 9x - \frac{3}{7}x \right) dx
\]

I will now calculate these integrals for you.The total area of the shaded region is approximately **887.40 square units**. This is the sum of the following areas:

1. The area between the parabola \( y = 52 - x^2 \) and the line \( y = 9x \), which is approximately **818.83 square units**.
2. The area between the lines \( y = 9x \) and \( y = \frac{3}{7}x \), which is approximately **68.57 square units**.

Would you like further details on any part of the solution or have any questions? Here are 5 related questions to explore:

1. How do you solve a quadratic equation like \( x^2 + 9x - 52 = 0 \)?
2. What are some techniques to find intersections between curves?
3. How do we interpret the geometric meaning of an integral in terms of area?
4. How can you visualize areas between curves using software?
5. What are the real-world applications of finding areas between curves?

**Tip:** Always check which curve is above the other when setting up integrals for area.

Find the area of the shaded region shown in the figure​ (which is not shown to​ scale). y = 52 - x^2, y = 9x, y = 3/7x

Learn how to calculate the area of the shaded region between the curves y = 52 - x^2, y = 9x, and y = 3/7x. This detailed solution involves solving quadratic equations and applying integral calculus to compute the area. Ideal for students in Grades 11-12.

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