# What is the arclength of #f(t) = (sqrt(t-2),t^2)# on #t in [2,3]#?

We see that:

Then:

And:

Combining these, we see that:

This can't be integrated by hand.

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The arc length of the curve defined by (f(t) = (\sqrt{t-2}, t^2)) on the interval ([2,3]) is given by the formula:

[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dt}\right)^2} dt]

where (a = 2) and (b = 3), and (\frac{dy}{dt}) is the derivative of (y) with respect to (t). Here, (y = t^2), so (\frac{dy}{dt} = 2t).

Plugging in the values, we get:

[L = \int_{2}^{3} \sqrt{1 + \left(\frac{d(t^2)}{dt}\right)^2} dt] [L = \int_{2}^{3} \sqrt{1 + (2t)^2} dt] [L = \int_{2}^{3} \sqrt{1 + 4t^2} dt]

This integral can be challenging to solve analytically. However, you can approximate the arc length using numerical methods or 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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