
Consider the curve C
 defined by the function f
 on the closed interval [0,4]
, where
f(x)=(2⋅(2⋅x+1)^3/2)/3


Compute the length of C
.
l(C)=
  


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?

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**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.

Consider the curve C defined by the function f on the closed interval [0,4], where f(x) = (2⋅(2⋅x+1)^3/2)/3. Compute the length of C. l(C) = ?

Learn how to compute the arc length of the curve f(x) = (2(2x + 1)^(3/2))/3 over the interval [0, 4]. This detailed solution includes derivative calculation, arc length formula application, and integration using substitution. Suitable for college-level calculus students.

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