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How do you find the arc length of the curve #y = 2 − x^2# from [0,1]?

Answer 1

Christopher Agresta

It is given by the integral #L=int_0^1sqrt(1+(dy/dx)^2)dx#

Hence #dy/dx=-2x# we have that

#L=int_0^1sqrt(1+4x^2)dx#

For the calculation of the integral see below

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Answer 2

Cameron Alexander

To find the arc length of the curve ( y = 2 - x^2 ) from ( x = 0 ) to ( x = 1 ), we use the arc length formula:

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

Where ( \frac{dy}{dx} ) represents the derivative of ( y ) with respect to ( x ).

First, find ( \frac{dy}{dx} ):

[ \frac{dy}{dx} = -2x ]

Now, plug ( \frac{dy}{dx} ) into the formula for arc length:

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

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

Now integrate ( \sqrt{1 + 4x^2} ) with respect to ( x ) from ( 0 ) to ( 1 ) to find the arc length.

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Answer from HIX Tutor

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.