Volumes with Cross Sections: Triangles and Semicircles · 具有横截面的体积：三角形和半圆
Same idea, different slice shapes
- The volume recipe $V=\int_a^b A(x)\,dx$ works for any cross-section shape — you just need $A(x)$.
- Beyond squares, the AP loves triangular and semicircular cross sections.
- The only new step is the right area formula for that shape, written in terms of the base.
- Get $A(x)$ from geometry, then integrate as before.
同样的想法,不同的切片形状
- 体积配方 $V=\int_a^b A(x)\,dx$ 对任何横截面形状都有效——你只需要 $A(x)$。
- 除了正方形,AP 还喜欢三角形和半圆形横截面。
- 唯一的新步骤是那个形状的正确面积公式,用底表示。
- 从几何得到 $A(x)$,再像之前那样积分。
Triangular cross sections
- For an equilateral triangle of side $s$: area $=\dfrac{\sqrt3}{4}s^2$.
- For a right isosceles triangle with legs $s$: area $=\dfrac12 s^2$; a general triangle: $\tfrac12\,\text{base}\times\text{height}$.
- Use the base length $s(x)$ (the gap the slice spans) in the correct triangle-area formula.
- $A(x)=(\text{shape constant})\cdot s(x)^2$, then integrate.
三角形横截面
- 对边长 $s$ 的等边三角形:面积 $=\dfrac{\sqrt3}{4}s^2$。
- 对直角边为 $s$ 的等腰直角三角形:面积 $=\dfrac12 s^2$;一般三角形:$\tfrac12\,\text{base}\times\text{height}$。
- 在正确的三角形面积公式中用底长 $s(x)$(切片跨过的间隙)。
- $A(x)=(\text{shape constant})\cdot s(x)^2$,再积分。
The base whose slices vary · 切片变化的底面
y = a·√x
The same base region gives different volumes depending on slice shape — only the area constant times $s(x)^2$ changes. · 相同的底面区域会因切片形状不同而产生不同的体积——仅当面积常数乘以$s(x)^2$时发生变化。
The area of an equilateral triangle with side $s$ is... · 边长为$s$的等边三角形面积为...
Equilateral triangle: $\frac{\sqrt3}{4}s^2$. · 等边三角形：$\frac{\sqrt3}{4}s^2$。
Match each cross-section shape to its area constant (times $s^2$). · 将每种横截面形状与其面积常数（乘以$s^2$）匹配。
Each area is its constant times the base squared. · 每个面积等于其常数乘以底边的平方。
Semicircular cross sections
- A semicircle with diameter $s$ has radius $\dfrac{s}{2}$, so its area is $\dfrac12\pi\Big(\dfrac{s}{2}\Big)^2=\dfrac{\pi}{8}s^2$.
- The base of the slice is the diameter, so halve it to get the radius before squaring.
- $A(x)=\dfrac{\pi}{8}\big(s(x)\big)^2$.
- Watch that factor: diameter $\to$ radius is a divide-by-two that's easy to skip.
半圆形横截面
- 直径为 $s$ 的半圆半径为 $\dfrac{s}{2}$,所以面积是 $\dfrac12\pi\Big(\dfrac{s}{2}\Big)^2=\dfrac{\pi}{8}s^2$。
- 切片的底是直径,所以平方前先减半得到半径。
- $A(x)=\dfrac{\pi}{8}\big(s(x)\big)^2$。
- 当心那个因子:直径 $\to$ 半径是一个容易漏掉的除以二。
A semicircular cross section has its diameter on the base $s(x)$. Its radius is... · 半圆形横截面的直径位于底面$s(x)$上。其半径为...
Base is the diameter, so radius is half. · 底面即为直径，所以半径是一半。
For a semicircle whose diameter is the base $s$, you must halve $s$ to get the radius before squaring. · 对于直径为底面$s$的半圆，必须在平方前将$s$减半以得到半径。
Radius $=s/2$; forgetting this is a common error. · 半径$=s/2$；忘记这一点是常见错误。
It's always "area in terms of the base"
- Every one of these is $A(x)=(\text{constant depending on shape})\times s(x)^2$.
- Square $\to 1$; equilateral triangle $\to \tfrac{\sqrt3}{4}$; semicircle $\to \tfrac{\pi}{8}$; right isosceles $\to \tfrac12$.
- Find the base $s(x)$ from the region, plug into the shape's area, and integrate.
- The calculus is identical; only the leading constant changes.
永远是"用底表示面积"
- 这些每一个都是 $A(x)=(\text{constant depending on shape})\times s(x)^2$。
- 正方形 $\to 1$;等边三角形 $\to \tfrac{\sqrt3}{4}$;半圆 $\to \tfrac{\pi}{8}$;等腰直角 $\to \tfrac12$。
- 从区域求底 $s(x)$,代入形状面积,再积分。
- 微积分完全相同;只有前面的常数在变。
Base under $y=\sqrt x$ on $[0,4]$, semicircular slices: $A(x)=\tfrac{\pi}{8}x$. Then $V=$ · $y=\sqrt x$在$[0,4]$上的底面，半圆形切片：$A(x)=\tfrac{\pi}{8}x$。然后$V=$
$\tfrac{\pi}{8}\int_0^4 x\,dx=\tfrac{\pi}{8}\cdot8=\pi$.
Every cross-section volume is $\int_a^b A(x)\,dx$, where $A(x)$ = (shape constant) $\times s(x)$ ____. · 每个横截面体积为$\int_a^b A(x)\,dx$，其中$A(x)$ = (形状常数) $\times s(x)$ ____。
Areas scale with the base squared. · 面积随底边的平方缩放。
For a semicircle, the slice's base is the diameter, so the radius is $\frac{s}{2}$ — don't square the whole base as if it were the radius. And match the triangle formula to the type stated (equilateral vs. right isosceles); they have different constants. The base $s(x)$ is still the gap the slice spans.
对半圆,切片的底是直径,所以半径是 $\frac{s}{2}$——别把整个底当作半径来平方。并把三角形公式匹配到所述的类型(等边 vs 等腰直角);它们的常数不同。底 $s(x)$ 仍是切片跨过的间隙。
Base is the region under $y=\sqrt x$ on $[0,4]$; cross sections are semicircles with diameter on the base.
- Diameter $s(x)=\sqrt x$, so radius $=\dfrac{\sqrt x}{2}$.
- $A(x)=\dfrac{\pi}{8}(\sqrt x)^2=\dfrac{\pi}{8}x$.
- $V=\displaystyle\int_0^4 \dfrac{\pi}{8}x\,dx=\dfrac{\pi}{8}\cdot 8 = \pi$.
底是 $[0,4]$ 上 $y=\sqrt x$ 下方的区域;横截面是直径在底上的半圆。
- 直径 $s(x)=\sqrt x$,所以半径 $=\dfrac{\sqrt x}{2}$。
- $A(x)=\dfrac{\pi}{8}(\sqrt x)^2=\dfrac{\pi}{8}x$。
- $V=\displaystyle\int_0^4 \dfrac{\pi}{8}x\,dx=\dfrac{\pi}{8}\cdot 8 = \pi$。
For triangular or semicircular cross sections, use $V=\int_a^b A(x)\,dx$ with the shape's area: equilateral triangle $\frac{\sqrt3}{4}s^2$, right isosceles $\frac12 s^2$, semicircle $\frac{\pi}{8}s^2$ (base $=$ diameter, so radius $=\frac s2$). Find the base $s(x)$ and integrate.
对三角形或半圆形横截面,用 $V=\int_a^b A(x)\,dx$ 配上形状面积:等边三角形 $\frac{\sqrt3}{4}s^2$、等腰直角 $\frac12 s^2$、半圆 $\frac{\pi}{8}s^2$(底 $=$ 直径,所以半径 $=\frac s2$)。求出底 $s(x)$ 再积分。