Question

Use the arc length formula to find the length of the curve y = 2 − x2 , 0 ≤ x ≤ 1. Check your answer by noting that the curve is part of a circle.

Answer 1

By using the arc length formula, we will see that the length of the curve is L = 1.48

How to use the arc length formula?

Here we have the curve:

y = 2 - x^2   with 0 ≤ x ≤ 1

And we want to find the length of the curve.

The arc length formula for a curve y in the interval [x₁, x₂] is given by:

$L =\int\limits^{x_2}_{x_1} {\sqrt{1 + \frac{dy}{dx}^2} } } \, dx$

For our curve, we have:

dy/dx = -2x

And the interval is [0, 1]

Replacing that we get:

$L =\int\limits^{1}_{0} {\sqrt{1 + (-2x)^2} } } \, dx\\\\L \int\limits^{1}_{0} {\sqrt{1 + 4x^2} } } \, dx$

This integral is not trivial, using a table you can see that this is equal to:

$L = (\frac{Arsinh(2x)}{4} + \frac{x*\sqrt{4x^2 + 1} }{2})$

evaluated from x = 1 to x = 0, when we do that, we will get:

$L = \frac{Arsinh(2) + 2*\sqrt{5} }{4} = 1.48$

That is the length of the curve.

If you want to learn more about curve length, you can read:

brainly.com/question/2005046