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We are tasked with evaluating the integral

\[
\int_0^4 y^3 \sqrt{9 - y^3} \, dy
\]

in terms of the **Beta function**. The general strategy here will be to use a substitution that transforms the integral into a form involving the Beta function.

### Step 1: Substitution
Let’s make the substitution:
\[
y^3 = 9 \sin^2(\theta)
\]
which implies that
\[
y = (9 \sin^2(\theta))^{1/3}
\]
and
\[
dy = \frac{3}{9 \sin(\theta) \cos(\theta)} .


Evaluate in terms of Beta function: ∫ from 0 to 4 y^3 √(9 - y^3) dy

Learn how to evaluate the integral ∫ from 0 to 4 y^3 √(9 - y^3) dy using substitution and expressing the result in terms of the Beta function. This solution involves key concepts from integral calculus and special functions.

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