To find the area of the region bounded by the curve given by the equation \( a^2 y^2 = x^2 (a^2 - x^2) \), we can follow these steps:
### Step 1: Rearranging the Equation
We start with the equation:
\[
a^2 y^2 = x^2 (a^2 - x^2)
\]
Dividing both sides by \( a^2 \) gives:
\[
y^2 = \frac{x^2 (a^2 - x^2)}{a^2}
\]
Taking the square root of both sides, we have:
\[
y = \pm \frac{x}{a} \sqrt{a^2 - x^2}
\]
### Step 2: Finding the Points of Intersection
Next, we need to find where this curve intersects the x-axis. The curve intersects the x-axis when \( y = 0 \):
\[
0 = \pm \frac{x}{a} \sqrt{a^2 - x^2}
\]
This occurs when \( x = 0 \) or \( x = a \). Thus, the points of intersection are \( (0, 0) \) and \( (a, 0) \).
### Step 3: Setting Up the Integral for Area
The area bounded by the curve can be calculated by integrating the positive part of the function from \( x = 0 \) to \( x = a \):
\[
\text{Area} = 4 \int_0^a y \, dx = 4 \int_0^a \frac{x}{a} \sqrt{a^2 - x^2} \, dx
\]
### Step 4: Simplifying the Integral
We can factor out \( \frac{1}{a} \):
\[
\text{Area} = \frac{4}{a} \int_0^a x \sqrt{a^2 - x^2} \, dx
\]
### Step 5: Using Substitution
To solve the integral, we use the substitution \( x = a \sin \theta \), which gives \( dx = a \cos \theta \, d\theta \). The limits change from \( x = 0 \) to \( x = a \) which corresponds to \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \):
\[
\int_0^a x \sqrt{a^2 - x^2} \, dx = \int_0^{\frac{\pi}{2}} (a \sin \theta) \sqrt{a^2 - (a \sin \theta)^2} \cdot a \cos \theta \, d\theta
\]
This simplifies to:
\[
= a^2 \int_0^{\frac{\pi}{2}} \sin \theta \cdot a \cos \theta \cdot \sqrt{a^2 (1 - \sin^2 \theta)} \, d\theta
\]
\[
= a^3 \int_0^{\frac{\pi}{2}} \sin \theta \cos^2 \theta \, d\theta
\]
### Step 6: Evaluating the Integral
The integral \( \int_0^{\frac{\pi}{2}} \sin \theta \cos^2 \theta \, d\theta \) can be solved using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \):
\[
= \int_0^{\frac{\pi}{2}} \sin \theta (1 - \sin^2 \theta) \, d\theta
\]
This can be evaluated to yield:
\[
= \frac{1}{3}
\]
### Step 7: Final Calculation of Area
Substituting back, we find:
\[
\text{Area} = \frac{4}{a} \cdot a^3 \cdot \frac{1}{3} = \frac{4a^2}{3}
\]
Thus, the area of the region bounded by the curve is:
\[
\text{Area} = \frac{4}{3} a^2
\]
