Determine the area under the curve `y=sqrt(a^(2)-x^(2))` included between the lines x =0 and x = a.

Explanation

To determine the area under the curve \( y = \sqrt{a^2 - x^2} \) between the lines \( x = 0 \) and \( x = a \), we can follow these steps:

### Step 1: Understand the Curve
The equation \( y = \sqrt{a^2 - x^2} \) represents the upper half of a circle with radius \( a \) centered at the origin. The full equation of the circle is \( x^2 + y^2 = a^2 \).

### Step 2: Set Up the Integral
To find the area under the curve from \( x = 0 \) to \( x = a \), we set up the integral:
\[ \text{Area} = \int_{0}^{a} \sqrt{a^2 - x^2} \, dx \]

### Step 3: Use the Integral Formula
The integral of \( \sqrt{a^2 - x^2} \) can be evaluated using the formula:
\[ \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) + C \]

### Step 4: Evaluate the Integral from 0 to a
Now we will evaluate the definite integral:
\[ \int_{0}^{a} \sqrt{a^2 - x^2} \, dx = \left[ \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) \right]_{0}^{a} \]

### Step 5: Substitute the Limits
Substituting the upper limit \( x = a \):
\[ = \left( \frac{a}{2} \sqrt{a^2 - a^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{a}{a}\right) \right) - \left( \frac{0}{2} \sqrt{a^2 - 0^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{0}{a}\right) \right) \]
\[ = \left( \frac{a}{2} \cdot 0 + \frac{a^2}{2} \cdot \frac{\pi}{2} \right) - \left( 0 + 0 \right) \]
\[ = \frac{a^2 \pi}{4} \]

### Step 6: Conclusion
Thus, the area under the curve \( y = \sqrt{a^2 - x^2} \) from \( x = 0 \) to \( x = a \) is:
\[ \text{Area} = \frac{\pi a^2}{4} \]