The animation linked to below is another holdover from older materials.
Play the video below to see an animation of the construction of the solid in space with base $R$ in the $x,y$-plane defined by $R=\{(x,y): x^2\leq y\leq 1\}$ and the property that cross-sections perpendicular to the $y$-axis are all squares. The areas of these cross-sections are given by $A(y)=(2\sqrt{y})^2 = 4y$, and so the total volume of the solid can be computed with the integral $\int_0^1 4y\,dy$.