The set of points (x,y) in the plane satisfying `x ^(2//5)+ |y| =1` form a curve enclosing a region of area `p/q` square units, when p and q are relatively prime positive intergers. Find `p-q.`

Answer

To solve the problem, we need to find the area enclosed by the curve defined by the equation \( x^{2/5} + |y| = 1 \). Let's break down the solution step by step.

### Step 1: Understand the Equation
The equation \( x^{2/5} + |y| = 1 \) can be separated into two cases based on the absolute value of \( y \):
1. \( y = 1 - x^{2/5} \) (for \( y \geq 0 \))
2. \( y = - (1 - x^{2/5}) \) (for \( y < 0 \))

### Step 2: Determine the Points of Intersection
We need to find the points where the curve intersects the axes:
- For \( y = 0 \):
\[ x^{2/5} = 1 \implies x = \pm 1 \]
- For \( x = 0 \):
\[ |y| = 1 \implies y = \pm 1 \]
Thus, the curve intersects the points \( (1, 0) \), \( (-1, 0) \), \( (0, 1) \), and \( (0, -1) \).

### Step 3: Sketch the Graph
The graph of the equations \( y = 1 - x^{2/5} \) and \( y = - (1 - x^{2/5}) \) will be symmetric about the y-axis. The curves will resemble a "flattened" parabola, opening downwards and upwards, respectively.

### Step 4: Set Up the Integral for Area Calculation
To find the area enclosed by the curves, we can calculate the area from \( x = -1 \) to \( x = 1 \). The total area \( A \) can be expressed as:
\[ A = 2 \int_{0}^{1} (1 - x^{2/5}) \, dx \]
This accounts for the area above the x-axis and below the curve.

### Step 5: Calculate the Integral
Now, we compute the integral:
\[ A = 2 \int_{0}^{1} (1 - x^{2/5}) \, dx = 2 \left[ \int_{0}^{1} 1 \, dx - \int_{0}^{1} x^{2/5} \, dx \right] \]
Calculating each integral:
1. \( \int_{0}^{1} 1 \, dx = [x]_{0}^{1} = 1 \)
2. \( \int_{0}^{1} x^{2/5} \, dx = \left[ \frac{5}{7} x^{7/5} \right]_{0}^{1} = \frac{5}{7} \)

Putting it all together:
\[ A = 2 \left( 1 - \frac{5}{7} \right) = 2 \left( \frac{2}{7} \right) = \frac{4}{7} \]

### Step 6: Identify \( p \) and \( q \)
The area \( A \) can be expressed as \( \frac{p}{q} \) where \( p = 4 \) and \( q = 7 \). Since 4 and 7 are relatively prime, we can now find \( p - q \):
\[ p - q = 4 - 7 = -3 \]

### Final Answer
Thus, the value of \( p - q \) is \( -3 \).