Volume with Washer Method: Revolving Around Other Axes AB & BC

Revolving about $y = k$ makes both radii distances from that line, $|f(x) - k|$ and $|g(x) - k|$, and the larger one is $R$. Adjusting only one of the two is the characteristic error here, and it produces a well-formed integral with no internal signal that anything is wrong, so the guard is to write both radii in the same form on the same line before proceeding. When the axis lies beyond the region the boundaries trade places: for $y = 2x$ above $y = x^{2}$ on $[0, 2]$ about $y = 5$, the parabola is four units from the axis at $x = 1$ against the line's three, so $R = 5 - x^{2}$ and $r = 5 - 2x$ and the volume is $\frac{136\pi}{15}$.

Keeping the 8.11 assignment gives the negative of that, which is the clearest available signal that the swap was missed. Three checks close the gap: both radii carry the same $k$ in the same form, $R \geq r$ at an interior test point, and the integrand vanishes wherever the two curves meet, since the washer degenerates to a circle there. Vertical axes add 8.10's requirements on top, so the variable, both conversions and the assignment of $R$ and $r$ all have to be right at once. Across 8.7 to 8.12 the integral has never changed; only the shape of $A$ and the length feeding it have.