The area bounded by the curve `xy^2=a^2(a−x)` and y-axis is

Explanation

To find the area bounded by the curve \( xy^2 = a^2(a - x) \) and the y-axis, we can follow these steps:

### Step 1: Rewrite the equation
The given equation is \( xy^2 = a^2(a - x) \). We can express \( y^2 \) in terms of \( x \):
\[ y^2 = \frac{a^2(a - x)}{x} \]
This shows how \( y \) depends on \( x \).

### Step 2: Determine the limits of integration
The curve intersects the x-axis when \( y = 0 \). Setting \( y^2 = 0 \) gives:
\[ a^2(a - x) = 0 \]
This implies \( x = a \) (since \( a^2 \neq 0 \)). The area we are interested in is bounded by the y-axis (where \( x = 0 \)) and the curve from \( x = 0 \) to \( x = a \).

### Step 3: Set up the integral for area
The area \( A \) can be calculated using the integral:
\[ A = \int_0^a y \, dx \]
To find \( y \), we take the square root of \( y^2 \):
\[ y = \sqrt{\frac{a^2(a - x)}{x}} = \frac{a\sqrt{a - x}}{\sqrt{x}} \]
Thus, the area becomes:
\[ A = \int_0^a \frac{a\sqrt{a - x}}{\sqrt{x}} \, dx \]

### Step 4: Use substitution to simplify the integral
We can use the substitution \( x = a \sin^2 \theta \). Then, \( dx = 2a \sin \theta \cos \theta \, d\theta \) and the limits change from \( x = 0 \) to \( x = a \) which corresponds to \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \).

### Step 5: Substitute and simplify the integral
Substituting \( x = a \sin^2 \theta \) into the area integral:
\[ A = \int_0^{\frac{\pi}{2}} \frac{a\sqrt{a - a \sin^2 \theta}}{\sqrt{a \sin^2 \theta}} \cdot 2a \sin \theta \cos \theta \, d\theta \]
This simplifies to:
\[ A = 2a^2 \int_0^{\frac{\pi}{2}} \frac{\cos \theta}{\sin \theta} \cdot \sin \theta \cos \theta \, d\theta = 2a^2 \int_0^{\frac{\pi}{2}} \cos^2 \theta \, d\theta \]

### Step 6: Evaluate the integral
Using the identity \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \):
\[ A = 2a^2 \int_0^{\frac{\pi}{2}} \frac{1 + \cos 2\theta}{2} \, d\theta = a^2 \left[ \theta + \frac{\sin 2\theta}{2} \right]_0^{\frac{\pi}{2}} = a^2 \left[ \frac{\pi}{2} + 0 - 0 \right] = \frac{\pi a^2}{2} \]

### Step 7: Final area calculation
Since the curve is symmetrical about the x-axis, we multiply the area by 2:
\[ \text{Total Area} = 2 \cdot \frac{\pi a^2}{2} = \pi a^2 \]

### Final Answer
The area bounded by the curve and the y-axis is:
\[ \boxed{\pi a^2} \text{ square units} \]