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The given problem involves a parameterized curve \( C \) defined by the coordinates \((x, y, z)\) with a parameter \( t \), where:

\[
x = t^2 - 8a, \quad y = t^2 + 2at, \quad z = t.
\]

We are also given a constant \( a \), and the integral to solve:

\[
I = \int_C (z^2 + 16z - x) \, ds,
\]

where \( 0 \leq t \leq 1 \).

### Step 1: Express \( ds \) in terms of \( t \)

We have:

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

1. Calculate \( \frac{dx}{dt} \):
   \[
   \frac{dx}{dt} = \frac{d}{dt}(t^2 - 8a) = 2t.
   \]

2. Calculate \( \frac{dy}{dt} \):
   \[
   \frac{dy}{dt} = \frac{d}{dt}(t^2 + 2at) = 2t + 2a.
   \]

3. Calculate \( \frac{dz}{dt} \):
   \[
   \frac{dz}{dt} = \frac{d}{dt}(t) = 1.
   \]

Thus,

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

### Step 2: Express the integrand in terms of \( t \)

We are given \( z = t \). Therefore:

\[
z^2 = t^2, \quad 16z = 16t.
\]

The integrand \( z^2 + 16z - x \) becomes:

\[
t^2 + 16t - (t^2 - 8a) = t^2 + 16t - t^2 + 8a = 16t + 8a.
\]

### Step 3: Formulate the integral \( I \)

The integral \( I \) can now be expressed as:

\[
I = \int_0^1 (16t + 8a) \, \sqrt{4t^2 + (2t + 2a)^2 + 1} \, dt.
\]

### Step 4: Simplify \( \sqrt{4t^2 + (2t + 2a)^2 + 1} \)

Expanding \( (2t + 2a)^2 \):

\[
(2t + 2a)^2 = 4t^2 + 8at + 4a^2.
\]

Thus,

\[
\sqrt{4t^2 + 4t^2 + 8at + 4a^2 + 1} = \sqrt{8t^2 + 8at + 4a^2 + 1}.
\]

### Final step: Evaluate or interpret the integral based on further simplifications

We have set up the integral with all terms in terms of \( t \), which allows for numerical evaluation or further algebraic simplification. If you'd like, we can proceed to explicitly evaluate this integral with the known bounds.

Would you like more details or further steps to compute the solution? Let me know!

Here are five follow-up questions based on this explanation:
1. How do you calculate \( ds \) in general for a parametric curve?
2. Why do we substitute the expressions for \( z^2 \), \( 16z \), and \( x \) in terms of \( t \)?
3. What are some strategies to simplify integrands in line integrals?
4. How would you interpret the physical or geometric meaning of the integral \( I \) if the curve represents a path?
5. Can this problem be solved more easily if we approximate \( ds \) for small \( t \) values?

**Tip:** When dealing with parameterized curves, it is crucial to carefully calculate the differentials \( dx \), \( dy \), and \( dz \) before formulating \( ds \) to avoid mistakes.

Points (x, y, z) on the curve C are given by x = t^2 - 8a, y = t^2 + 2at, and z = t, where a is a positive constant. Which one is the solution of the integral I = ∫C (z^2 + 16z - x) ds for 0 ≤ t ≤ 1?

Learn how to solve the line integral I = ∫C (z^2 + 16z - x) ds for a parametric curve defined by x = t^2 - 8a, y = t^2 + 2at, and z = t. This problem covers key calculus concepts such as parametric curves, differentials, and line integrals, suitable for advanced calculus students.

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