The problem asks to find the area $A$ bounded by the curves $x = y^2 - 7$ and $x = 8 - y^2$.

To solve this, we will:

1. Find the points of intersection between the two curves by equating them: $y^2 - 7 = 8 - y^2$
2. Solve this equation to find the limits of integration (the values of $y$).
3. Set up the integral for the area, which is the difference between the two functions $x = 8 - y^2$ (the right curve) and $x = y^2 - 7$ (the left curve).
4. Evaluate the integral to find the area.

Let's begin by solving for the points of intersection.

$y^2 - 7 = 8 - y^2$

This simplifies to:

$2y^2 = 15$

$y^2 = \frac{15}{2}$

$y = \pm \sqrt{\frac{15}{2}} = \pm \frac{\sqrt{30}}{2}$

So, the limits of integration for $y$ are $-\frac{\sqrt{30}}{2}$ to $\frac{\sqrt{30}}{2}$.

Now, we set up the integral for the area. The area between the curves is:

$A = \int_{-\frac{\sqrt{30}}{2}}^{\frac{\sqrt{30}}{2}} \left[(8 - y^2) - (y^2 - 7)\right] \, dy$

Simplifying the integrand:

$A = \int_{-\frac{\sqrt{30}}{2}}^{\frac{\sqrt{30}}{2}} (15 - 2y^2) \, dy$

I will now calculate this integral to find the area.The area $A$ bounded by the curves $x = y^2 - 7$ and $x = 8 - y^2$ is approximately $54.7723$ square units, rounded to four decimal places.

Would you like further details or have any questions?

Here are some related questions to consider:

1. How would the area change if one of the curves shifted vertically or horizontally?
2. What are the geometric interpretations of these curves?
3. Could you find the area between these curves using a different method, like parametric equations?
4. How would the integral setup change if the curves were rotated?
5. Can this process be applied to curves of higher degree?

Tip: Always double-check the points of intersection when solving for the area between curves, as incorrect limits can lead to errors in your final result.