The arc of the curve xy² + x - y = 0 between the middle

The arc of the curve $xy^2 + x - y = 0$ between the middle inflection point and the point with the largest abscissa rotates about the $y$-axis. Find the volume of the resulting solid.

We identify the arc's two endpoints from calculus conditions, then integrate using the disk method.

Step 1: Express $x$ in terms of $y$

\[x(y^2+1) = y \implies x = \frac{y}{y^2+1}\]

Step 2: Find the point of largest abscissa

\[\frac{dx}{dy} = \frac{1-y^2}{(y^2+1)^2} = 0 \implies y=\pm1\]

At $y=1$: $x=\frac12$ (the largest value of $x$ in the first quadrant branch).

Step 3: Find the middle inflection point

\[\frac{d^2x}{dy^2} = \frac{-2y(3-y^2)}{(y^2+1)^3} = 0 \implies y=0 \text{ or } y=\pm\sqrt3\]

The middle inflection point (nearest the origin) is at $y=0$, giving $x=0$ — i.e. the point $(0,0)$.

Step 4: Set up the disk method

The arc runs from $(0,0)$ to $\left(\frac12,1\right)$, i.e. $y\in[0,1]$. Rotating about the $y$-axis:

\[A(y) = \pi x^2 = \frac{\pi y^2}{(y^2+1)^2}\]

Step 5: Decompose the integrand

\[\frac{y^2}{(y^2+1)^2} = \frac{(y^2+1)-1}{(y^2+1)^2} = \frac{1}{y^2+1}-\frac{1}{(y^2+1)^2}\]

Step 6: Integrate each piece

\[\int_0^1\frac{1}{y^2+1}\,dy = [\arctan y]_0^1 = \frac\pi4\]

\[\int_0^1\frac{1}{(y^2+1)^2}\,dy = \left[\frac12\left(\arctan y+\frac{y}{y^2+1}\right)\right]_0^1 = \frac12\left(\frac\pi4+\frac12\right) = \frac\pi8+\frac14\]

Step 7: Combine

\[\int_0^1\frac{y^2}{(y^2+1)^2}\,dy = \frac\pi4-\left(\frac\pi8+\frac14\right) = \frac\pi8-\frac14\]

\[V = \pi\left(\frac\pi8-\frac14\right) = \frac{\pi^2}{8}-\frac\pi4 = \frac{\pi(\pi-2)}{8}\]

\[\boxed{V = \frac{\pi(\pi-2)}{8} \text{ cubic units}}\]

Step 8: Numerical value

\[V = \frac{\pi(\pi-2)}{8} \approx 0.448 \text{ cubic units}\]

Summary

\[\boxed{V = \frac{\pi(\pi-2)}{8} \approx 0.448 \text{ cubic units}}\]

Key insight: The arc's two defining endpoints come from two different derivative conditions — a first-derivative extremum (largest abscissa) and a second-derivative inflection — and the resulting integral only needs the simple decomposition $\frac{y^2}{(y^2+1)^2}=\frac{1}{y^2+1}-\frac{1}{(y^2+1)^2}$ to become elementary.