Compute the double integral: $$ \int_{0}^{4} \int_{0}^{\frac{x}{2}} x y^{3} d x d y $$ (5 Marks)

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To compute the double integral

$$
\int_{0}^{4} \int_{0}^{\frac{x}{2}} x y^{3} \, dy \, dx,
$$

we will follow these steps:

1. **Identify the order of integration**: The integral is set up to integrate with respect to $$y$$ first and then $$x$$.

2. **Integrate with respect to $$y$$**: We will first compute the inner integral

$$
\int_{0}^{\frac{x}{2}} x y^{3} \, dy.
$$

Since $$x$$ is treated as a constant during this integration, we can factor it out:

$$
= x \int_{0}^{\frac{x}{2}} y^{3} \, dy.
$$

Now, we compute the integral of $$y^{3}$$:

$$
\int y^{3} \, dy = \frac{y^{4}}{4}.
$$

Evaluating this from $$0$$ to $$\frac{x}{2}$$:

$$
\left[ \frac{y^{4}}{4} \right]_{0}^{\frac{x}{2}} = \frac{1}{4} \left( \frac{x}{2} \right)^{4} = \frac{1}{4} \cdot \frac{x^{4}}{16} = \frac{x^{4}}{64}.
$$

Thus, the inner integral becomes:

$$
\int_{0}^{\frac{x}{2}} x y^{3} \, dy = x \cdot \frac{x^{4}}{64} = \frac{x^{5}}{64}.
$$

3. **Integrate with respect to $$x$$**: Now we compute the outer integral:

$$
\int_{0}^{4} \frac{x^{5}}{64} \, dx.
$$

Factoring out the constant $$\frac{1}{64}$$:

$$
= \frac{1}{64} \int_{0}^{4} x^{5} \, dx.
$$

Now, we compute the integral of $$x^{5}$$:

$$
\int x^{5} \, dx = \frac{x^{6}}{6}.
$$

Evaluating this from $$0$$ to $$4$$:

$$
\left[ \frac{x^{6}}{6} \right]_{0}^{4} = \frac{4^{6}}{6} = \frac{4096}{6} = \frac{2048}{3}.
$$

Thus, the outer integral becomes:

$$
\frac{1}{64} \cdot \frac{2048}{3} = \frac{2048}{192} = \frac{64}{3}.
$$

4. **Final Result**: Therefore, the value of the double integral is

$$
\boxed{\frac{64}{3}}.
$$
The value of the double integral is $$ \frac{64}{3} $$.

Compute the double integral:
(5 Marks)

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Compute the double integral: (5 Marks)