`int_(sqrt(2)//3)^(sqrt(3)//3)(1)/(sqrt(4-9x^(2)))dx=`

Options

A. `(pi)/(36)`
B. `(pi)/(9)`
C. `(pi)/(12)`
D. `(pi)/(4)`

Answer

To solve the integral

\[ \int_{\frac{\sqrt{2}}{3}}^{\frac{\sqrt{3}}{3}} \frac{1}{\sqrt{4 - 9x^2}} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Integral
First, we rewrite the expression under the square root:

\[ \sqrt{4 - 9x^2} = \sqrt{4(1 - \frac{9}{4}x^2)} = 2\sqrt{1 - \frac{9}{4}x^2}. \]

Thus, the integral becomes:

\[ \int_{\frac{\sqrt{2}}{3}}^{\frac{\sqrt{3}}{3}} \frac{1}{2\sqrt{1 - \frac{9}{4}x^2}} \, dx = \frac{1}{2} \int_{\frac{\sqrt{2}}{3}}^{\frac{\sqrt{3}}{3}} \frac{1}{\sqrt{1 - \frac{9}{4}x^2}} \, dx. \]

### Step 2: Use a Substitution
Let \( a^2 = \frac{4}{9} \) so that \( a = \frac{2}{3} \). We can use the formula for the integral:

\[ \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C. \]

Thus, we have:

\[ \int \frac{1}{\sqrt{1 - \frac{9}{4}x^2}} \, dx = \frac{2}{3} \sin^{-1}\left(\frac{3x}{2}\right) + C. \]

### Step 3: Evaluate the Integral
Now we can evaluate the definite integral:

\[ \frac{1}{2} \left[ \frac{2}{3} \sin^{-1}\left(\frac{3x}{2}\right) \right]_{\frac{\sqrt{2}}{3}}^{\frac{\sqrt{3}}{3}} = \frac{1}{3} \left[ \sin^{-1}\left(\frac{3 \cdot \frac{\sqrt{3}}{3}}{2}\right) - \sin^{-1}\left(\frac{3 \cdot \frac{\sqrt{2}}{3}}{2}\right) \right]. \]

This simplifies to:

\[ \frac{1}{3} \left[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) - \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) \right]. \]

### Step 4: Substitute Values
We know:

\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}, \quad \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}. \]

So we substitute these values into our expression:

\[ \frac{1}{3} \left[ \frac{\pi}{3} - \frac{\pi}{4} \right]. \]

### Step 5: Simplify the Result
Now we need to simplify:

\[ \frac{1}{3} \left[ \frac{4\pi}{12} - \frac{3\pi}{12} \right] = \frac{1}{3} \left[ \frac{\pi}{12} \right] = \frac{\pi}{36}. \]

### Final Answer
Thus, the value of the integral is:

\[ \int_{\frac{\sqrt{2}}{3}}^{\frac{\sqrt{3}}{3}} \frac{1}{\sqrt{4 - 9x^2}} \, dx = \frac{\pi}{36}. \]