How do you find the arc length of the curve #y=x^2/2# over the interval [0, 1]?
Computer used for integration and numerical solution
... and then maybe a hyperbolic double angle formula]
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To find the arc length of the curve ( y = \frac{x^2}{2} ) over the interval ([0, 1]), you can use the formula for arc length:
[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]
Here, ( a = 0 ) and ( b = 1 ). First, find ( \frac{dy}{dx} ), then plug it into the formula and integrate over the given interval.

Find ( \frac{dy}{dx} ): [ \frac{dy}{dx} = x ]

Substitute ( \frac{dy}{dx} = x ) into the arc length formula: [ L = \int_{0}^{1} \sqrt{1 + x^2} , dx ]

Integrate ( \sqrt{1 + x^2} ) from 0 to 1 to find the arc length.
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To find the arc length of the curve (y = \frac{x^2}{2}) over the interval ([0, 1]), we use the formula for arc length of a curve given by:
[ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} , dx ]
Where (f(x)) is the given function, and (a) and (b) are the lower and upper limits of integration, respectively.
For the given function (y = \frac{x^2}{2}), we first find its derivative:
[ f'(x) = x ]
Now, we substitute this derivative into the formula:
[ L = \int_{0}^{1} \sqrt{1 + x^2} , dx ]
This integral represents the arc length of the curve (y = \frac{x^2}{2}) over the interval ([0, 1]). We evaluate this integral to find the arc length.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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