`int_(0)^(2//3)sqrt(4-9x^(2))dx=`

Options

A. `pi`
B. `(pi)/(2)`
C. `(pi)/(3)`
D. `(pi)/(4)`

Answer

To solve the definite integral \( \int_{0}^{\frac{2}{3}} \sqrt{4 - 9x^2} \, dx \), we can follow these steps:

### Step 1: Factor out the constant
We start by factoring out the constant from the square root:
\[ \int_{0}^{\frac{2}{3}} \sqrt{4 - 9x^2} \, dx = \int_{0}^{\frac{2}{3}} \sqrt{9\left(\frac{4}{9} - x^2\right)} \, dx \]
This simplifies to:
\[ = \int_{0}^{\frac{2}{3}} 3\sqrt{\frac{4}{9} - x^2} \, dx \]

### Step 2: Factor out the constant from the integral
Now we can factor out the constant \(3\):
\[ = 3 \int_{0}^{\frac{2}{3}} \sqrt{\frac{4}{9} - x^2} \, dx \]

### Step 3: Use the substitution
Next, we can use the property of integrals. We recognize that this integral resembles the form \( \int \sqrt{a^2 - x^2} \, dx \). Here, \( a = \frac{2}{3} \):
\[ = 3 \left[ \frac{x}{2} \sqrt{\frac{4}{9} - x^2} + \frac{4}{9} \cdot \frac{1}{2} \sin^{-1}\left(\frac{3x}{2}\right) \right]_{0}^{\frac{2}{3}} \]

### Step 4: Evaluate the first part
Now we evaluate the first part of the integral at the limits:
\[ = 3 \left[ \frac{\frac{2}{3}}{2} \sqrt{\frac{4}{9} - \left(\frac{2}{3}\right)^2} - 0 \right] \]
Calculating this gives:
\[ = 3 \left[ \frac{1}{3} \sqrt{\frac{4}{9} - \frac{4}{9}} \right] = 3 \left[ \frac{1}{3} \cdot 0 \right] = 0 \]

### Step 5: Evaluate the second part
Now we evaluate the second part:
\[ = 3 \left[ 0 + \frac{4}{9} \cdot \frac{1}{2} \sin^{-1}\left(1\right) - 0 \right] \]
Since \( \sin^{-1}(1) = \frac{\pi}{2} \):
\[ = 3 \left[ \frac{4}{9} \cdot \frac{1}{2} \cdot \frac{\pi}{2} \right] = 3 \cdot \frac{4\pi}{36} = \frac{4\pi}{12} = \frac{\pi}{3} \]

### Final Result
Thus, the value of the definite integral is:
\[ \int_{0}^{\frac{2}{3}} \sqrt{4 - 9x^2} \, dx = \frac{\pi}{9} \]