`int_(a)^(b) sqrt((x-a)(b-x)) dx, (b gt a)` is equal to

Options

A. `(pi(b-a)^(2))/8`
B. `(pi(b+a)^(2))/8`
C. `(b-a)^(2)`
D. `(b+a)^(2)`

Explanation

To solve the integral \( I = \int_{a}^{b} \sqrt{(x-a)(b-x)} \, dx \), where \( b > a \), we can use a trigonometric substitution. Here’s a step-by-step solution:

### Step 1: Substitution
Let’s use the substitution:
\[ x = a \cos^2 \theta + b \sin^2 \theta \]
This substitution helps in transforming the integral into a more manageable form.

### Step 2: Calculate \( dx \)
Differentiating \( x \) with respect to \( \theta \):
\[ dx = \frac{d}{d\theta}(a \cos^2 \theta + b \sin^2 \theta) = -2a \cos \theta \sin \theta + 2b \sin \theta \cos \theta = (2b - 2a) \sin \theta \cos \theta \, d\theta \]
Thus, we can express \( dx \) as:
\[ dx = (b - a) \sin(2\theta) \, d\theta \]

### Step 3: Change the limits of integration
When \( x = a \):
\[ a = a \cos^2 \theta + b \sin^2 \theta \implies \sin^2 \theta = 0 \implies \theta = 0 \]
When \( x = b \):
\[ b = a \cos^2 \theta + b \sin^2 \theta \implies \cos^2 \theta = 0 \implies \theta = \frac{\pi}{2} \]
So the limits change from \( x = a \) to \( x = b \) into \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \).

### Step 4: Substitute into the integral
Now we substitute \( x \) and \( dx \) into the integral:
\[ I = \int_{0}^{\frac{\pi}{2}} \sqrt{(a \cos^2 \theta + b \sin^2 \theta - a)(b - (a \cos^2 \theta + b \sin^2 \theta))} \cdot (b - a) \sin(2\theta) \, d\theta \]

### Step 5: Simplify the integrand
We simplify the terms inside the square root:
\[ x - a = b \sin^2 \theta \quad \text{and} \quad b - x = a \cos^2 \theta \]
Thus, we have:
\[ I = (b - a) \int_{0}^{\frac{\pi}{2}} \sqrt{b \sin^2 \theta \cdot a \cos^2 \theta} \cdot \sin(2\theta) \, d\theta \]
This simplifies to:
\[ I = (b - a) \int_{0}^{\frac{\pi}{2}} \sqrt{ab} \sin(2\theta) \cdot \sin(2\theta) \, d\theta \]
\[ = (b - a) \sqrt{ab} \int_{0}^{\frac{\pi}{2}} \sin^2(2\theta) \, d\theta \]

### Step 6: Evaluate the integral
We know:
\[ \int \sin^2(2\theta) \, d\theta = \frac{1}{2} \left(\theta - \frac{1}{4} \sin(4\theta)\right) \]
Evaluating from \( 0 \) to \( \frac{\pi}{2} \):
\[ = \frac{1}{2} \left(\frac{\pi}{2} - 0\right) = \frac{\pi}{4} \]

### Step 7: Final result
Putting it all together:
\[ I = (b - a) \sqrt{ab} \cdot \frac{\pi}{4} \]
Thus, the final result is:
\[ I = \frac{\pi}{4} (b - a) \sqrt{ab} \]