Question

106. Arc length Find the length of the curve y = (x / 2) * sqrt(3 - $$x^2) + (3 / 2) * sin^(-1)(x / $$sqrt(3)) from x = 0 to x = 1.

Accepted Answer

Identify the formula for the arc length of a curve defined by a function $$ y = f(x) $$ from $$ x = a  to$$ $$ x = b $$:

$$
L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
$$ Given the function:

$$
y = \frac{x}{2} \sqrt{3 - x^2} + \frac{3}{2} \sin^{-1}\left(\frac{x}{\sqrt{3}}\right)

we $$need to find its derivative $$ \frac{dy}{dx} . $$Use the product rule and chain rule for the first term, and the derivative of the inverse sine function for the second term. Compute $$ \frac{dy}{dx} $$ step-by-step:

- For the first term $$ \frac{x}{2} \sqrt{3 - x^2} $$, write it as $$ \frac{x}{2} (3 - x^2)^{1/2} $$ and apply the product rule:

$$
\frac{d}{dx} \left( \frac{x}{2} (3 - x^2)^{1/2} \right) = \frac{1}{2} (3 - x^2)^{1/2} + \frac{x}{2} \cdot \frac{d}{dx} (3 - x^2)^{1/2}

- $$For the derivative of $$ (3 - x^2)^{1/2} $$, use the chain rule:

$$
\frac{d}{dx} (3 - x^2)^{1/2} = \frac{1}{2} (3 - x^2)^{-1/2} \cdot (-2x) = -\frac{x}{(3 - x^2)^{1/2}}

- $$For the second term $$ \frac{3}{2} \sin^{-1}\left(\frac{x}{\sqrt{3}}\right) $$, use the derivative of $$ \sin^{-1}(u) $$:

$$
\frac{d}{dx} \sin^{-1}(u) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx}
$$

where $$ u = \frac{x}{\sqrt{3}} $$, so $$ \frac{du}{dx} = \frac{1}{\sqrt{3}} . $$After finding $$ \frac{dy}{dx} $$, square it to get $$ \left(\frac{dy}{dx}\right)^2 . $$Then, substitute into the arc length integral formula:

$$
L = \int_0^1 \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
$$ Set up the integral with the expression inside the square root simplified as much as possible. Finally, evaluate the integral either by analytical methods if possible or by numerical approximation to find the length of the curve from $$ x = 0  to$$ $$ x = 1 .$$

106. Arc length Find the length of the curve y = (x / 2) * sqrt(3 - x^2) + (3 / 2) * sin^(-1)(x / sqrt(3)) from x = 0 to x = 1.

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Key Concepts

Arc Length Formula

Derivative of Composite and Inverse Functions

Integration Techniques