# How do you find the length of the curve defined by #f(x) = x^2# on the x-interval (0, 3)?

The formula for the arc length for a curve is:

So, we need to find the derivative, and then square it:

Plugging it in:

Assuming you are in your first year of Calculus, this is not something you might have been taught how to do. So maybe you should evaluate this on your calculator or on Wolfram Alpha numerically.

But, if you have heard of trigonometric substitution, that is what I would do.

So you now have the substituted integral (ignoring the bounds for now):

To do this difficult integral, we have to use integration by parts...

Thus:

Using the substitutions from before:

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To find the length of the curve defined by f(x) = x^2 on the x-interval (0, 3), you can use the arc length formula:

[L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx]

where f'(x) is the derivative of the function f(x) with respect to x.

For the given function f(x) = x^2, its derivative is f'(x) = 2x.

So, substituting into the formula:

[L = \int_{0}^{3} \sqrt{1 + (2x)^2} , dx]

[L = \int_{0}^{3} \sqrt{1 + 4x^2} , dx]

You can then evaluate this integral to find the length of the curve.

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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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