By using the arc length formula, we will see that the length of the curve is L = 1.48
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:
[tex]L =\int\limits^{x_2}_{x_1} {\sqrt{1 + \frac{dy}{dx}^2} } } \, dx[/tex]
For our curve, we have:
dy/dx = -2x
And the interval is [0, 1]
Replacing that we get:
[tex]L =\int\limits^{1}_{0} {\sqrt{1 + (-2x)^2} } } \, dx\\\\L \int\limits^{1}_{0} {\sqrt{1 + 4x^2} } } \, dx\\[/tex]
This integral is not trivial, using a table you can see that this is equal to:
[tex]L = (\frac{Arsinh(2x)}{4} + \frac{x*\sqrt{4x^2 + 1} }{2})[/tex]
evaluated from x = 1 to x = 0, when we do that, we will get:
[tex]L = \frac{Arsinh(2) + 2*\sqrt{5} }{4} = 1.48[/tex]
That is the length of the curve.
If you want to learn more about curve length, you can read:
https://brainly.com/question/2005046
