` int _(0)^(a) sqrt(a^(2) - x^(2))` dx is equal to

Options

A. `pia^(2)`
B. `1/2 pia^(2)`
C. `1/3 pia^(2)`
D. `1/4 pia^(2)`

Explanation

To solve the integral \( \int_0^a \sqrt{a^2 - x^2} \, dx \), we can follow these steps:

### Step 1: Identify the Integral
We need to evaluate the integral:
\[ I = \int_0^a \sqrt{a^2 - x^2} \, dx \]

### Step 2: Use the General Result
The integral \( \int \sqrt{a^2 - x^2} \, dx \) has a known result:
\[ \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) + C \]
where \( C \) is the constant of integration.

### Step 3: Apply the Limits
Now we will apply the limits from \( 0 \) to \( a \):
\[ I = \left[ \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) \right]_0^a \]

### Step 4: Evaluate at the Upper Limit \( x = a \)
Substituting \( x = a \):
\[ I = \frac{a}{2} \sqrt{a^2 - a^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{a}{a}\right) \]
\[ = \frac{a}{2} \cdot 0 + \frac{a^2}{2} \cdot \frac{\pi}{2} = \frac{a^2 \pi}{4} \]

### Step 5: Evaluate at the Lower Limit \( x = 0 \)
Substituting \( x = 0 \):
\[ I = \frac{0}{2} \sqrt{a^2 - 0^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{0}{a}\right) \]
\[ = 0 + \frac{a^2}{2} \cdot 0 = 0 \]

### Step 6: Combine the Results
Now, we combine the results from the upper and lower limits:
\[ I = \left( \frac{a^2 \pi}{4} \right) - 0 = \frac{a^2 \pi}{4} \]

### Final Result
Thus, the value of the integral is:
\[ \int_0^a \sqrt{a^2 - x^2} \, dx = \frac{a^2 \pi}{4} \]

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