Representative rectangle

Draw it on the sketch before writing any integral. A rectangle of thickness $dx$ stands vertically, so its height is a difference of $y$ values and every boundary has to be written as a function of $x$. A rectangle of thickness $dy$ lies horizontally, so the boundaries must first be solved for $x$ in terms of $y$.

Rotating that rectangle is what builds the solid. Swung about an axis it is perpendicular to, it sweeps a circular plate whose radii are measured from the axis to each end of the rectangle. When the rectangle sits on the axis the near radius is zero and the plate is a solid disk of radius equal to the height; when a gap separates the rectangle from the axis the plate is a washer with outer radius $R$ the distance to the far end and inner radius $r$ the distance to the near end, and the height is $R-r$, not a radius. Swung about an axis it is parallel to, it wraps into a hollow tube whose radius is its distance from the axis and whose height stays its height.