The area of the region enclosed by the curve `|y|=-(1-|x|)^2+5,` is

Options

A. `8/3 (7+5sqrt(5))` sq units
B. `2/3 (7+5sqrt(5))` sq units
C. `2/3 (5sqrt(5)-7)` sq units
D. None of these

Answer

To find the area of the region enclosed by the curve \( |y| = -(1 - |x|)^2 + 5 \), we will follow these steps:

### Step 1: Understand the equation
The equation \( |y| = -(1 - |x|)^2 + 5 \) can be rewritten as:
\[ y = -(1 - |x|)^2 + 5 \quad \text{and} \quad y = (1 - |x|)^2 - 5 \]
This means we will have two curves, one for positive \( y \) and one for negative \( y \).

### Step 2: Find the points of intersection
To find the points where the curve intersects the x-axis, set \( y = 0 \):
\[ 0 = -(1 - |x|)^2 + 5 \]
Rearranging gives:
\[ (1 - |x|)^2 = 5 \]
Taking square roots:
\[ 1 - |x| = \pm \sqrt{5} \]
This leads to two cases:
1. \( 1 - |x| = \sqrt{5} \) → \( |x| = 1 - \sqrt{5} \) (not valid since \( \sqrt{5} > 1 \))
2. \( 1 - |x| = -\sqrt{5} \) → \( |x| = 1 + \sqrt{5} \)

Thus, the points of intersection are \( x = \pm(1 + \sqrt{5}) \).

### Step 3: Determine the area
The area enclosed by the curve can be calculated by integrating the function \( y = -(1 - |x|)^2 + 5 \) from \( - (1 + \sqrt{5}) \) to \( 1 + \sqrt{5} \).

Since the curve is symmetric about the y-axis, we can calculate the area for \( x \) from \( 0 \) to \( 1 + \sqrt{5} \) and then double it:
\[ \text{Area} = 2 \int_0^{1 + \sqrt{5}} (-(1 - x)^2 + 5) \, dx \]

### Step 4: Simplify the integral
First, simplify the integrand:
\[ -(1 - x)^2 + 5 = - (1 - 2x + x^2) + 5 = -1 + 2x - x^2 + 5 = 2x - x^2 + 4 \]

### Step 5: Set up the integral
Now we can set up the integral:
\[ \text{Area} = 2 \int_0^{1 + \sqrt{5}} (2x - x^2 + 4) \, dx \]

### Step 6: Calculate the integral
Calculating the integral:
\[ \int (2x - x^2 + 4) \, dx = x^2 - \frac{x^3}{3} + 4x \]
Evaluating from \( 0 \) to \( 1 + \sqrt{5} \):
\[ \left[ (1 + \sqrt{5})^2 - \frac{(1 + \sqrt{5})^3}{3} + 4(1 + \sqrt{5}) \right] - \left[ 0 \right] \]

### Step 7: Calculate the definite integral
Now we calculate:
1. \( (1 + \sqrt{5})^2 = 1 + 2\sqrt{5} + 5 = 6 + 2\sqrt{5} \)
2. \( (1 + \sqrt{5})^3 = 1 + 3\sqrt{5} + 3 \cdot 5 + 5\sqrt{5} = 16 + 8\sqrt{5} \)

Putting these into the integral:
\[ \text{Area} = 2 \left[ (6 + 2\sqrt{5}) - \frac{(16 + 8\sqrt{5})}{3} + 4(1 + \sqrt{5}) \right] \]

### Step 8: Final simplification
Combine and simplify:
\[ = 2 \left[ (6 + 2\sqrt{5}) - \frac{16}{3} - \frac{8\sqrt{5}}{3} + 4 + 4\sqrt{5} \right] \]
\[ = 2 \left[ \left( 6 + 4 - \frac{16}{3} \right) + \left( 2\sqrt{5} + 4\sqrt{5} - \frac{8\sqrt{5}}{3} \right) \right] \]
Calculating the constants and the coefficients of \( \sqrt{5} \) gives the final area.

### Final Result
The area of the region enclosed by the curve is \( \frac{28}{3} - \frac{25\sqrt{5}}{3} \).

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