To solve the integral

\[
\int_0^{\frac{2}{3}} -6x^3 \sqrt{4 - 9x^2} \, dx
\]

we will follow these steps:

### Step 1: Simplify the expression

We are given the integrand as \( -6x^3 \sqrt{4 - 9x^2} \). This doesn't simplify easily, but we can try a trigonometric substitution to handle the square root term.

### Step 2: Trigonometric substitution

The term inside the square root is of the form \( a^2 - b^2x^2 \), where \( a = 2 \) and \( b = 3 \). This suggests using the substitution:

\[
x = \frac{2}{3} \sin(\theta)
\]
which leads to:

\[
dx = \frac{2}{3} \cos(\theta) d\theta
\]

Now, substitute \( x = \frac{2}{3} \sin(\theta) \) into the integrand:

\[
\sqrt{4 - 9x^2} = \sqrt{4 - 9 \left(\frac{4}{9} \sin^2(\theta)\right)} = \sqrt{4(1 - \sin^2(\theta))} = 2 \cos(\theta)
\]

### Step 3: Apply the substitution to the entire integrand

The integrand becomes:

\[
-6 \left(\frac{2}{3} \sin(\theta)\right)^3 2 \cos(\theta) \cdot \frac{2}{3} \cos(\theta) d\theta
\]

Simplify the expression step by step.

Let me calculate this transformation and proceed with solving the integral.It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like me to guide you through the solution manually?

Here are some relative questions for further exploration:
1. How can we handle integrals involving square roots?
2. What is the typical process for applying trigonometric substitutions in integrals?
3. Can we generalize the substitution \(x = \frac{a}{b} \sin(\theta)\) for other integrals?
4. How do you deal with integrals where cubic powers of \(x\) are involved?
5. Can you use numerical methods to approximate the value of such integrals?

**Tip:** Trigonometric substitution is especially useful for integrals involving expressions like \(a^2 - b^2x^2\), reducing them to simpler trigonometric forms.

Learn how to solve the integral of -6x^3 sqrt(4 - 9x^2) using trigonometric substitution. This problem covers advanced calculus concepts including integral transformations and square root simplification. Suitable for undergraduate students familiar with calculus.

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