To compute the length of the curve $C$, which is given by the function $f(x) = \frac{2 \cdot (2x + 1)^{3/2}}{3}$, over the closed interval $[0, 4]$, we use the formula for the arc length of a curve defined by $y = f(x)$:

$l(C) = \int_{a}^{b} \sqrt{1 + \left( f'(x) \right)^2} \, dx$

Step 1: Find the derivative of $f(x)$, i.e., $f'(x)$.

The function $f(x) = \frac{2 \cdot (2x + 1)^{3/2}}{3}$. To find the derivative, apply the chain rule:

$f'(x) = \frac{2}{3} \cdot \frac{3}{2} \cdot (2x + 1)^{1/2} \cdot 2$

Simplifying:

$f'(x) = 2 \cdot (2x + 1)^{1/2}$

Step 2: Plug $f'(x)$ into the arc length formula.

Now, the formula becomes:

$l(C) = \int_{0}^{4} \sqrt{1 + \left( 2 \cdot (2x + 1)^{1/2} \right)^2} \, dx$

Simplify inside the square root:

$l(C) = \int_{0}^{4} \sqrt{1 + 4 \cdot (2x + 1)} \, dx$

$l(C) = \int_{0}^{4} \sqrt{4 \cdot (2x + 1)} \, dx$

$l(C) = \int_{0}^{4} 2 \cdot \sqrt{2x + 1} \, dx$

Step 3: Compute the integral.

We now compute:

$l(C) = 2 \int_{0}^{4} \sqrt{2x + 1} \, dx$

We can solve this integral by substitution. Let $u = 2x + 1$, so $du = 2dx$, or $dx = \frac{du}{2}$.

When $x = 0$, $u = 1$. When $x = 4$, $u = 9$.

Thus, the integral becomes:

$l(C) = 2 \int_{1}^{9} \sqrt{u} \cdot \frac{du}{2}$

Simplify:

$l(C) = \int_{1}^{9} \sqrt{u} \, du$

$l(C) = \int_{1}^{9} u^{1/2} \, du$

Step 4: Integrate.

The integral of $u^{1/2}$ is:

$\frac{2}{3} u^{3/2}$

Now, evaluate this from $1$ to $9$:

$l(C) = \left[ \frac{2}{3} u^{3/2} \right]_1^9$

$l(C) = \frac{2}{3} \left( 9^{3/2} - 1^{3/2} \right)$

$l(C) = \frac{2}{3} \left( 27 - 1 \right)$

$l(C) = \frac{2}{3} \times 26 = \frac{52}{3}$

Thus, the length of the curve $C$ is:

$l(C) = \frac{52}{3}$

Final Answer:

$l(C) = \frac{52}{3} \approx 17.33$

Would you like more details, or have any questions?

---

Relative questions:

1. How is the arc length formula derived for curves?
2. Can arc length be applied in parametric equations?
3. What is the effect of scaling the function $f(x)$ on the arc length?
4. How would you compute the arc length if $f(x)$ were piecewise?
5. What happens if the interval changes from $[0, 4]$ to $[1, 5]$?

Tip: Always check if simplifying expressions inside the integral makes solving easier.