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The problem presented is asking to evaluate a line integral of the form:

\[
\int_C y^3 \, ds
\]

where the curve \( C \) is given parametrically as \( x = t^3 \) and \( y = t \), with \( 0 \leq t \leq 5 \).

### Steps to solve the line integral:

1. **Calculate the differential of arc length \( ds \):**
   The formula for \( ds \) for a curve parametrized by \( t \) is:

   \[
   ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt
   \]

   Given that \( x = t^3 \) and \( y = t \), let's calculate the derivatives:

   \[
   \frac{dx}{dt} = 3t^2, \quad \frac{dy}{dt} = 1
   \]

   Therefore,

   \[
   ds = \sqrt{(3t^2)^2 + (1)^2} \, dt = \sqrt{9t^4 + 1} \, dt
   \]

2. **Substitute the values into the integral:**

   The integrand is \( y^3 \), and since \( y = t \), we have \( y^3 = t^3 \). Thus, the line integral becomes:

   \[
   \int_0^5 t^3 \sqrt{9t^4 + 1} \, dt
   \]

3. **Evaluate the integral:**

   This is a non-trivial integral that likely requires numerical integration. I will now compute this integral.

Let me calculate that for you.The correct value of the line integral is approximately:

\[
7814.56
\]

This result differs slightly from the provided value \( 7808.46 \), possibly due to rounding or a calculation error in the original attempt.

Would you like any additional details on the steps, or do you have further questions?

Here are five related questions to consider:
1. How does the formula for \( ds \) change for different types of parameterized curves?
2. How do you calculate the differential arc length for a curve in three dimensions?
3. Can you explain the physical interpretation of a line integral in vector fields?
4. What other numerical methods could be used for evaluating more complex integrals?
5. How do line integrals relate to work done by a force field?

**Tip:** When working with parametric curves, always check the bounds of the parameter carefully to ensure you're integrating over the correct portion of the curve.

Evaluate the line integral: ∫_C y^3 ds, where the curve C is parametrized as x = t^3, y = t, and 0 ≤ t ≤ 5.

Learn how to evaluate a line integral of the form ∫_C y^3 ds, where C is a parametric curve defined by x = t^3, y = t for 0 ≤ t ≤ 5. This example includes key concepts of parametric curves, arc length, and numerical integration techniques.

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