Key Concepts and Formulas

• Area between curves: If $f(x) \ge g(x)$ on $[a, b]$, the area is $\int_a^b (f(x) - g(x)) \, dx$.
• Symmetry: Exploit symmetry to simplify the area calculation.
• Equation of a circle: $x^2 + y^2 = r^2$ represents a circle centered at $(0,0)$ with radius $r$.

Step-by-Step Solution

Step 1: Analyze the given equations and identify the region

We are given $|y| = 1 - x^2$ and $x^2 + y^2 = 1$. The first equation represents two parabolas: $y = 1 - x^2$ for $y \ge 0$, and $y = x^2 - 1$ for $y < 0$. The second equation is a circle with radius 1 centered at the origin. We need to find the area enclosed between these curves.

Step 2: Find the points of intersection

To find the points of intersection, we substitute $y^2 = (1 - x^2)^2$ into $x^2 + y^2 = 1$: $x^2 + (1 - x^2)^2 = 1$ $x^2 + 1 - 2x^2 + x^4 = 1$ $x^4 - x^2 = 0$ $x^2(x^2 - 1) = 0$ So, $x = 0, x = 1, x = -1$. When $x = 0$, $y^2 = 1$, so $y = \pm 1$. The points are $(0, 1)$ and $(0, -1)$. When $x = 1$, $y^2 = 0$, so $y = 0$. The point is $(1, 0)$. When $x = -1$, $y^2 = 0$, so $y = 0$. The point is $(-1, 0)$.

The intersection points are $(0, 1)$, $(0, -1)$, $(1, 0)$, and $(-1, 0)$.

Step 3: Exploit symmetry to simplify the area calculation

The region is symmetric about both the x-axis and the y-axis. Therefore, we can find the area in the first quadrant and multiply by 4. In the first quadrant, $x$ ranges from 0 to 1. The upper curve is $y = 1 - x^2$ (from $|y| = 1 - x^2$) and the lower curve is $y = \sqrt{1 - x^2}$ (from $x^2 + y^2 = 1$). Note that the circle is above the parabola in the first quadrant.

Step 4: Set up and evaluate the integral

The area in the first quadrant is given by: $A_1 = \int_0^1 (\sqrt{1 - x^2} - (1 - x^2)) \, dx$ The total area $\alpha$ is $4A_1$: $\alpha = 4 \int_0^1 (\sqrt{1 - x^2} - (1 - x^2)) \, dx = 4 \int_0^1 \sqrt{1 - x^2} \, dx - 4 \int_0^1 (1 - x^2) \, dx$ We know that $\int_0^1 \sqrt{1 - x^2} \, dx$ represents the area of a quarter circle with radius 1, so $\int_0^1 \sqrt{1 - x^2} \, dx = \frac{\pi}{4}$. The second integral is: $\int_0^1 (1 - x^2) \, dx = \left[x - \frac{x^3}{3}\right]_0^1 = 1 - \frac{1}{3} = \frac{2}{3}$ Therefore, $\alpha = 4 \left(\frac{\pi}{4}\right) - 4 \left(\frac{2}{3}\right) = \pi - \frac{8}{3}$

Step 5: Calculate $9\alpha$ and find $\beta$ and $\gamma$

We are given that $9\alpha = \beta \pi + \gamma$. $9\alpha = 9\left(\pi - \frac{8}{3}\right) = 9\pi - 24$ So, $\beta = 9$ and $\gamma = -24$.

Step 6: Calculate $|\beta - \gamma|$

We need to find $|\beta - \gamma| = |9 - (-24)| = |9 + 24| = |33| = 33$. Wait! This is not the provided answer.

Step 7: Re-examine the setup of the integral

In the first quadrant, the circle $y = \sqrt{1-x^2}$ is above the parabola $y = 1-x^2$. We set up the integral as: $A_1 = \int_0^1 (\sqrt{1 - x^2} - (1 - x^2)) \, dx$ This is correct. Let's check the integration again. $4 \int_0^1 (\sqrt{1 - x^2} - (1 - x^2)) \, dx = 4 \left( \frac{\pi}{4} - \frac{2}{3} \right) = \pi - \frac{8}{3}$ Then $9 \alpha = 9(\pi - \frac{8}{3}) = 9\pi - 24$. So $\beta = 9$ and $\gamma = -24$.

$|\beta - \gamma| = |9 - (-24)| = |9 + 24| = 33$. There must be an error in the question or options.

Let's look at the region enclosed by $|y| = 1 - x^2$ and $x^2 + y^2 = 1$. The area of the lens is $\alpha$. $9\alpha = \beta \pi + \gamma$. Find $|\beta - \gamma|$.

The area of the circle is $\pi$. The area inside $|y| = 1 - x^2$ is $2 \int_{-1}^1 (1 - x^2) dx = 2 [x - \frac{x^3}{3}]_{-1}^1 = 2[(1 - \frac{1}{3}) - (-1 + \frac{1}{3})] = 2[\frac{2}{3} + \frac{2}{3}] = \frac{8}{3}$.

Then $\alpha = \frac{8}{3} - \pi$. This must be wrong.

We calculated $\alpha = \pi - \frac{8}{3}$. So $9\alpha = 9\pi - 24$. $\beta = 9$ and $\gamma = -24$. Thus $|\beta - \gamma| = |9 - (-24)| = 33$.

I suspect the correct area should be the area inside the lens minus the area of the circle inside the lens. In the first quadrant the lens is below the circle. So, $\alpha = 4(\frac{\pi}{4} - \frac{2}{3}) = \pi - \frac{8}{3}$

$9\alpha = 9\pi - 24$ so $\beta = 9$ and $\gamma = -24$. Then $|\beta - \gamma| = 33$.

Since the answer MUST be 15, as stated in the original prompt, let's work backwards. $|\beta - \gamma| = 15$. $9\alpha = \beta \pi + \gamma$. If $\beta - \gamma = 15$, then $\beta = \gamma + 15$. $9\alpha = (\gamma + 15)\pi + \gamma = \gamma\pi + 15\pi + \gamma$. If $\gamma - \beta = 15$, then $\gamma = \beta + 15$. $9\alpha = \beta\pi + \beta + 15 = \beta(\pi + 1) + 15$.

Since $\alpha = \pi - \frac{8}{3}$, then $9\alpha = 9\pi - 24$. Comparing $9\pi - 24$ to $\beta\pi + \gamma$, we get $\beta = 9$ and $\gamma = -24$. Then $|\beta - \gamma| = |9 - (-24)| = 33$.

Let's assume $\alpha = \frac{5\pi}{3}$. Then $9\alpha = 15\pi$. So $9\pi - 24$ must be $15\pi + \gamma$. $-6\pi = 24 + \gamma$

There seems to be an error in the question itself. Given the curves and the definition of $\alpha$, $9\alpha = 9\pi - 24$, and $|\beta - \gamma| = 33$.

Common Mistakes & Tips

• Be careful about which curve is "above" the other when setting up the integral. Drawing a diagram is crucial.
• Remember to multiply by the appropriate factor when exploiting symmetry.
• Double-check your integration and algebraic manipulations.

Summary

We found the area enclosed between $|y| = 1 - x^2$ and $x^2 + y^2 = 1$ by exploiting symmetry and setting up a definite integral. We calculated the area in the first quadrant and multiplied it by 4 to get the total area. We then calculated $9\alpha$ and identified $\beta$ and $\gamma$. Finally, we computed $|\beta - \gamma|$. Despite arriving at 33, the problem states the answer should be 15. There is likely an error in the question. However, based on the described area, the correct computation leads to a value of 33.

Final Answer

The correct answer, based on the given curves and the described area, is 33. However, the provided correct answer is 15. Therefore, there is an error in the question. If we assume the computation is correct and the options are wrong, then the correct answer should be 33. If, however, we force the answer to be 15, it would require a different problem statement.

Given the discrepancy, I will assume there is an error in the options or the prompt and will proceed with the derived value of 33. Since none of the options match, I cannot choose one. The final answer is \boxed{33}.