The area enclosed by the curve `y^2 +x^4=x^2` is

Options

A. (a) `(2)/(3)`
B. (b) `(4)/(3)`
C. (c) `(8)/(3)`
D. (d) `(16)/(3)`

Answer

To find the area enclosed by the curve \( y^2 + x^4 = x^2 \), we can follow these steps:

### Step 1: Rearranging the equation
We start with the equation of the curve:
\[ y^2 + x^4 = x^2 \]
Rearranging gives:
\[ y^2 = x^2 - x^4 \]

### Step 2: Analyzing symmetry
The equation \( y^2 = x^2 - x^4 \) indicates that the curve is symmetric about the x-axis because if \( (x, y) \) is a point on the curve, then \( (x, -y) \) is also a point on the curve. We can also see that the equation is symmetric about the y-axis since replacing \( x \) with \( -x \) does not change the equation.

### Step 3: Finding the limits of integration
To find the area, we need to determine the points where the curve intersects the x-axis. Setting \( y = 0 \) in the equation gives:
\[ 0 = x^2 - x^4 \implies x^2(x^2 - 1) = 0 \]
This results in:
\[ x^2 = 0 \quad \text{or} \quad x^2 = 1 \implies x = 0, \, 1, \, -1 \]
Thus, the limits of integration are from \( -1 \) to \( 1 \).

### Step 4: Setting up the integral for area
Since the curve is symmetric about the x-axis, we can calculate the area above the x-axis and then double it:
\[ \text{Area} = 2 \int_0^1 y \, dx \]
Substituting \( y = \sqrt{x^2 - x^4} \):
\[ \text{Area} = 2 \int_0^1 \sqrt{x^2 - x^4} \, dx \]

### Step 5: Simplifying the integral
We can factor out \( x^2 \) from the square root:
\[ \sqrt{x^2 - x^4} = \sqrt{x^2(1 - x^2)} = x \sqrt{1 - x^2} \]
Thus, the area becomes:
\[ \text{Area} = 2 \int_0^1 x \sqrt{1 - x^2} \, dx \]

### Step 6: Substituting for integration
Let \( t = 1 - x^2 \), then \( dt = -2x \, dx \) or \( dx = -\frac{dt}{2x} \). When \( x = 0 \), \( t = 1 \) and when \( x = 1 \), \( t = 0 \):
\[ \text{Area} = 2 \int_1^0 x \sqrt{t} \left(-\frac{dt}{2x}\right) = \int_0^1 \sqrt{t} \, dt \]

### Step 7: Evaluating the integral
Now we can evaluate the integral:
\[ \int_0^1 \sqrt{t} \, dt = \left[ \frac{t^{3/2}}{3/2} \right]_0^1 = \frac{2}{3} \]
Thus:
\[ \text{Area} = 2 \cdot \frac{2}{3} = \frac{4}{3} \]

### Conclusion
The area enclosed by the curve \( y^2 + x^4 = x^2 \) is:
\[ \boxed{\frac{4}{3}} \text{ square units} \]