The length of a parametric curve given by $x(t)$ and $y(t)$ over an interval $a \leq t \leq b$ is determined by the arc length formula:

$L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

Given functions:

$x(t) = t^5 + 6$ $y(t) = 7\cos(5t^3)$

Compute the derivatives:

1. Derivative of $x(t)$: $\frac{dx}{dt} = \frac{d}{dt} \left( t^5 + 6 \right) = 5t^4$

2. Derivative of $y(t)$: Using the chain rule: $\frac{dy}{dt} = 7 \cdot \left(-\sin(5t^3) \right) \cdot \left( 15t^2 \right)$ $= -105t^2 \sin(5t^3)$

Arc length integral:

$L = \int_{3}^{9} \sqrt{(5t^4)^2 + (-105t^2 \sin(5t^3))^2} \, dt$

$L = \int_{3}^{9} \sqrt{25t^8 + 11025t^4 \sin^2(5t^3)} \, dt$

This integral represents the exact arc length of the given parametric curve from $t=3$ to $t=9$.

---

Would you like a step-by-step numerical approximation for this integral?

Here are five related questions:

1. How do you approximate an arc length integral numerically?
2. What is the geometric meaning of arc length in parametric curves?
3. How does the arc length formula change for different parameterizations?
4. How would you set up an integral for surface area if this curve were rotated about an axis?
5. Can we express the integral in terms of an alternative parameter?

Tip: When computing arc length, check if the function allows simplifications before integrating!