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Please help me find the Arc length with the given equation

To find the arc length \( L \) of a curve described by a parametric function, we can use the formula:

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

From your image, it appears we have:
\[
L = \int_0^3 \sqrt{(3)^2 + (3t^2 + 2)^2} \, dt
\]

To proceed, let’s calculate each part step-by-step:

### Step 1: Simplify the Integrand
The integrand inside the square root is:
\[
(3)^2 + (3t^2 + 2)^2
\]
Calculating each term:
- \( (3)^2 = 9 \)
- \( (3t^2 + 2)^2 = 9t^4 + 12t^2 + 4 \)

So the expression under the square root becomes:
\[
9 + 9t^4 + 12t^2 + 4 = 9t^4 + 12t^2 + 13
\]

### Step 2: Set Up the Integral
The arc length integral now is:
\[
L = \int_0^3 \sqrt{9t^4 + 12t^2 + 13} \, dt
\]

This integral may require advanced techniques (e.g., numerical integration or specific substitutions) to evaluate. However, if you need a numerical answer, it can be computed with a calculator or software capable of handling definite integrals.

Would you like to proceed with a numerical solution, or do you need further steps detailed? 

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### Related Questions

1. How do we find the arc length of a curve defined in polar coordinates?
2. What is the formula for arc length in a 3-dimensional parametric curve?
3. How do you handle integrals with complex expressions under the square root?
4. What are some common techniques for evaluating difficult integrals?
5. How does changing the limits of integration affect the arc length calculation?

### Tip
When dealing with arc length, simplifying the integrand before attempting to integrate often saves time, especially with complex expressions.

This guide walks through the calculation of arc length for a parametric equation from 0 to 3, using the arc length formula in calculus. Learn step-by-step how to simplify and set up the integral for an accurate arc length calculation. Suitable for Grades 11-12.

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