# How do you find the length of the curve for #y=x^2# for (0, 3)?

Arc Length

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To find the length of the curve (y = x^2) from (x = 0) to (x = 3), we use the arc length formula:

[L = \int_{x_1}^{x_2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx]

For (y = x^2), (\frac{dy}{dx} = 2x). Substituting into the formula and integrating from (x = 0) to (x = 3):

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

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

This integral may be evaluated using trigonometric substitution. Let (x = \frac{1}{2}\tan(\theta)), then (dx = \frac{1}{2}\sec^2(\theta) , d\theta). Substituting:

[L = \int_{0}^{\arctan(6)} \sqrt{1 + \tan^2(\theta)} \cdot \frac{1}{2}\sec^2(\theta) , d\theta]

[L = \int_{0}^{\arctan(6)} \sqrt{\sec^2(\theta)} \cdot \frac{1}{2}\sec^2(\theta) , d\theta]

[L = \int_{0}^{\arctan(6)} \frac{1}{2}\sec^3(\theta) , d\theta]

[L = \frac{1}{2} \int_{0}^{\arctan(6)} \sec^3(\theta) , d\theta]

This integral can be evaluated using integration techniques such as integration by parts or tables of integrals. Once evaluated, you'll obtain 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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