Volumes with Cross Sections: Squares and Rectangles AB & BC

A solid sliced perpendicular to an axis has volume $\int_a^b A(x)\,dx$, where $A(x)$ is the area of one cross section: an area times a thickness is a volume. The shape of the slice comes from the problem and its base length comes from the region, as $s = y_{\text{top}} - y_{\text{bottom}}$, which is 8.4's integrand reused. Taking $s$ from a single curve assumes the lower boundary is the axis, which is true in most introductory examples and false in general; checking $s$ at one interior point and at the ends, where it should vanish, catches it.

For square cross sections $A = s^{2}$, so the base between $y = x$ and $y = x^{2}$ on $[0, 1]$ gives $V = \int_0^1 (x - x^{2})^{2}\,dx = \frac{1}{30}$. Failing to square gives $\frac{1}{6}$, the area of the base, and squaring the terms separately gives $\frac{2}{15}$ by way of the false identity $(a - b)^{2} = a^{2} - b^{2}$. A rectangle whose height is $k$ times its base has $A = k s^{2}$, while one of fixed height $3$ has $A = 3s$. Every remaining topic in the unit is this same integral with a different $A$, discs and washers included.