Concept preview

What you will be able to do

1. 01Set up the volume of a solid with known cross sections as $V=\displaystyle\int_a^b A(x)\,dx$, where $A(x)$ is the cross-sectional area written entirely in terms of the base variable.
2. 02Read the side length of a square or rectangular cross section off the base region as $s(x)=(\text{top curve})-(\text{bottom curve})$, then build the area $A=s^2$ (square) or $A=s\cdot h$ (rectangle).
3. 03Evaluate the resulting definite integral to a clean value, and recognise the classic slip of forgetting to SQUARE the side before integrating.

Picture a two-dimensional region lying flat in the $xy$-plane — call it the BASE of the solid. Now, at each value of $x$ between $a$ and $b$, we erect a flat slice standing straight up out of the plane, perpendicular to the $x$-axis. The shape of that slice (square, rectangle, triangle, semicircle) is given to you in the problem, and its size is controlled by the base: as the base gets wider or narrower, the slice grows or shrinks with it. The solid is the union of all these slices packed side by side from $x=a$ to $x=b$.