Verified answer

y = (2 - x^2)^(1/2)

y^2 = 2 - x^2

2y * dy = -2x * dx

y * dy = -x * dx

dy/dx = -x / y

Arclength is the integral of:

sqrt(1 + (dy/dx)^2) * dx

sqrt(1 + (-x/y)^2) * dx =>

sqrt(1 + x^2 / y^2) * dx =>

sqrt((y^2 + x^2) / y^2) * dx =>

sqrt((x^2 + 2 - x^2) / (2 - x^2)) * dx =>

sqrt(2 / (2 - x^2)) * dx =>

sqrt(2) * dx / sqrt(2 - x^2)

x = sqrt(2) * sin(t)

dx = sqrt(2) * cos(t) * dt

sqrt(2) * sqrt(2) * cos(t) * dt / sqrt(2 - 2sin(t)^2) =>

2 * cos(t) * dt / sqrt(2 * (1 - sin(t)^2)) =>

2 * cos(t) * dt / sqrt(2 * cos(t)^2)) =>

2 * cos(t) * dt / (sqrt(2) * cos(t)) =>

sqrt(2) * dt

Integrate

sqrt(2) * t + C

Solve for t

x = sqrt(2) * sin(t)

x / sqrt(2) = sin(t)

t = arcsin(x / sqrt(2))

sqrt(2) * arcsin(x / sqrt(2)) + C

From 0 to 1

sqrt(2) * arcsin(1 / sqrt(2)) - sqrt(2) * arcsin(0 / sqrt(2)) =>

sqrt(2) * arcsin(sqrt(2)/2) - sqrt(2) * arcsin(0) =>

sqrt(2) * (pi/4) - sqrt(2) * 0 =>

(sqrt(2)/4) * pi

Use The Arc Length Formula

dy/dx = -x/(2 - x^2)^(1/2).

So, √[1 + (dy/dx)^2]

= √[1 + x^2/(2 - x^2)]

= √[((2 - x^2) + x^2)/(2 - x^2)]

= √2 / √(2 - x^2).

Hence, the arc length equals

∫(x = 0 to 1) √2 dx / √(2 - x^2)

= √2 arcsin(x/√2) {for x = 0 to 1}

= π√2 / 4.

Double check:

Note that the arc (with radius √2) is between 0 and 45 degrees (that is 1/8-th of the full circle).

So, the length equals (1/8) 2π * √2 = π√2 / 4.

I hope this helps!

y = √(2 - x^2)

dy/dx = - x /√(2 - x^2)

(dy/dx)^2 = x^2 /(2 - x^2)

1 + (dy/dx)^2 = ( 2 - x^2 + x^2) / (2 - x^2) = 2 / (2 - x^2) = 1 / (1 - (x^2/2))

SQRT [ 1 + (dy/dx)^2 ] = 1 /√( 1 - (x/√2)^2 )

arc length. s = ∫ dx /√( 1 - (x/√2)^2 ) from 0 to 1

let x/√2 = sin(t) when x = 0, t = 0 and when x = 1, t = π/4

dx = √2 cos(t) dt

s = ∫√2 cos(t) dt /√( 1 - sin^2(t) ) from 0 to π/4

= ∫√2 dt from 0 to π/4

= √2 t from 0 to π/4

= √2π/4

The given curve is y = √(2 - x^2)

=> y^2 = 2 - x^2

=> x^2 + y^2 = 2

This is eqn of circle with radius √2

we are looking circumference of 1/8 of circle ( 0 to 2π is full circle and 0 to π/4 is 1/8 circle)

C = 1/8(2π√2) = 1/4(√2π)

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