How do you find the arc length of the curve #y = sqrt( 2 − x^2 )#, #0 ≤ x ≤ 1#?
I found:
I plotted the arc you need (a bit of a cirle) and found:
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To find the arc length of the curve y = √(2 − x^2) over the interval 0 ≤ x ≤ 1, you use the formula for arc length:
[ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} , dx ]
Where ( f(x) = \sqrt{2  x^2} ) is the function representing the curve.
 Find the derivative of ( f(x) ), denoted as ( f'(x) ).
 Plug the derivative into the arc length formula.
 Integrate the expression from the lower bound to the upper bound of the given interval.
Here are the steps:

Find ( f'(x) ) by differentiating ( f(x) = \sqrt{2  x^2} ). [ f'(x) = \frac{x}{\sqrt{2  x^2}} ]

Plug ( f'(x) ) into the arc length formula: [ L = \int_{0}^{1} \sqrt{1 + \left(\frac{x}{\sqrt{2  x^2}}\right)^2} , dx ]

Simplify the expression under the square root: [ L = \int_{0}^{1} \sqrt{1 + \frac{x^2}{2  x^2}} , dx ]

Integrate the expression: [ L = \int_{0}^{1} \sqrt{\frac{2  x^2 + x^2}{2  x^2}} , dx ] [ L = \int_{0}^{1} \sqrt{\frac{2}{2  x^2}} , dx ]

Use trigonometric substitution or other methods to evaluate the integral.
Once the integral is evaluated, you'll have the arc length of the curve over the specified interval.
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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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