# How do you determine the length of #x=3t^2#, #y=t^3+4t# for t is between [0,2]?

Hi, are you sure about the two parametric equations?

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To determine the length of the curve defined by (x = 3t^2) and (y = t^3 + 4t) for (t) between ([0, 2]), we use the formula for arc length of a parametric curve:

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

Where:

- (a) and (b) are the lower and upper bounds of (t), respectively (in this case, (a = 0) and (b = 2)).
- (\frac{dx}{dt}) and (\frac{dy}{dt}) are the derivatives of (x) and (y) with respect to (t), respectively.

First, find the derivatives of (x) and (y) with respect to (t):

[\frac{dx}{dt} = 6t] [\frac{dy}{dt} = 3t^2 + 4]

Now, plug these derivatives into the formula and integrate over the given range:

[L = \int_{0}^{2} \sqrt{(6t)^2 + (3t^2 + 4)^2} dt]

[L = \int_{0}^{2} \sqrt{36t^2 + 9t^4 + 24t^2 + 16} dt]

[L = \int_{0}^{2} \sqrt{9t^4 + 60t^2 + 16} dt]

This integral may not have a simple closed-form solution. It can be evaluated using numerical methods or specialized software.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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