Volumes with Cross Sections: Squares and Rectangles

Stacking up thin slabs into a solid

• Integration also finds volumes, not just areas — by adding up thin slices of a solid.
• Slice a solid perpendicular to an axis; each slice is a thin slab of some cross-sectional area $A(x)$.
• Add them up: $V=\displaystyle\int_a^b A(x)\,dx$ — the integral of the cross-sectional area.
• This lesson covers solids whose cross sections 横截面 are squares or rectangles.

The general volume formula

• If the cross section at $x$ has area $A(x)$ and thickness $dx$, its tiny volume is $A(x)\,dx$.
• Sum over the solid:
  $$V=\int_a^b A(x)\,dx$$
• Everything reduces to: write the cross-sectional area as a function of $x$, then integrate.
• The hard part is the geometry of one slice, not the calculus.

Square cross sections on a base region: add the slice areas A(x) along the x-axis

Square cross sections

• A common setup: cross sections are squares with base sitting on the region between two curves.
• If the base length is $s(x)$ (often $f(x)-g(x)$, the gap between curves), a square's area is $s(x)^2$.
• So $V=\displaystyle\int_a^b \big(s(x)\big)^2\,dx$.
• The base of each square spans the region; its area is base-squared.

Rectangular cross sections

• For rectangles, the area is base $\times$ height: $A(x)=s(x)\cdot h(x)$.
• The height might be a fixed multiple of the base, or given separately in the problem.
• $V=\displaystyle\int_a^b s(x)\,h(x)\,dx$.
• Read the problem carefully to get the height rule right.

Write from a volume of 8/3

• Cross sections perpendicular to the x-axis are squares.
• The side of each square is s(x) = x, from 0 to 2.