Volumes with Cross Sections: Squares and Rectangles · 具有横截面的体积：正方形和矩形
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| cross sections/krɒs ˈsekʃnz/ | 横截面 | héng jié miàn |
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.
把薄板叠成一个立体
- 积分也能求体积,不只是面积——通过把立体的薄片加起来。
- 把立体垂直于某轴切片;每片是一块薄板,有某个横截面积 $A(x)$。
- 把它们加起来:$V=\displaystyle\int_a^b A(x)\,dx$——横截面积的积分。
- 这一课讲横截面是正方形或矩形的立体。
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.
一般体积公式
- 若 $x$ 处的横截面积为 $A(x)$、厚为 $dx$,它的微小体积是 $A(x)\,dx$。
- 沿立体求和:$$V=\int_a^b A(x)\,dx$$
- 一切归结为:把横截面积写成 $x$ 的函数,再积分。
- 难点是单个切片的几何,而非微积分。
The volume of a solid with cross-sectional area $A(x)$ is... · 横截面面积为$A(x)$的立体体积为...
Add up slabs $A(x)\,dx$. · 累加薄片$A(x)\,dx$。
To use $V=\int_a^b A(x)\,dx$, the cross-section slices must be... · 要使用$V=\int_a^b A(x)\,dx$，横截面切片必须与...
Slices are perpendicular to the axis you integrate along. · 切片垂直于你进行积分的轴。
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.
正方形横截面
- 常见设置:横截面是正方形,底坐落在两曲线之间的区域上。
- 若底长为 $s(x)$(常是 $f(x)-g(x)$,曲线间的间隙),正方形面积是 $s(x)^2$。
- 所以 $V=\displaystyle\int_a^b \big(s(x)\big)^2\,dx$。
- 每个正方形的底跨过区域;它的面积是底的平方。
The base region under √x · √x 下方的底面区域
y = a·√x
Each square slice sits on the base; its side is the height $\sqrt{x}$, so its area is $(\sqrt{x})^2=x$. · 每个正方形切片置于底面上；其边长为$\sqrt{x}$，因此面积为$(\sqrt{x})^2=x$。
For square cross sections with base $s(x)$, the area is... · 对于底面为$s(x)$的正方形横截面，面积为...
A square of side $s$ has area $s^2$. · 边长为$s$的正方形面积为$s^2$。
The base length of a cross section is often the distance between the boundary curves, $f(x)-g(x)$. · 横截面的底边长度通常是边界曲线之间的距离，即$f(x)-g(x)$。
The base spans the region between the curves. · 底边跨越两条曲线之间的区域。
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.
矩形横截面
- 对矩形,面积是底 $\times$ 高:$A(x)=s(x)\cdot h(x)$。
- 高可能是底的固定倍数,或在题目中单独给出。
- $V=\displaystyle\int_a^b s(x)\,h(x)\,dx$。
- 仔细读题以搞对高的规则。
Base under $y=\sqrt x$ on $[0,4]$, square cross sections: $A(x)=x$. Find $V=\int_0^4 x\,dx$. · $y=\sqrt x$在$[0,4]$上的底面，正方形横截面：$A(x)=x$。求$V=\int_0^4 x\,dx$。
$\big[\tfrac{x^2}{2}\big]_0^4=8$.
For rectangular cross sections, area = base ____ height. · 对于矩形横截面，面积 = 底边 ____ 高。
$A=s\cdot h$ for a rectangle. · 矩形的$A=s\cdot h$。
The base length $s(x)$ is usually the distance between the boundary curves ($f(x)-g(x)$) or between a curve and an axis — figure out what spans the base before squaring. For squares, the area is $s(x)^2$ (don't forget to square the whole gap). Slices must be perpendicular to the axis you integrate along.
底长 $s(x)$ 通常是边界曲线之间的距离($f(x)-g(x)$)或曲线与轴之间的距离——平方前先弄清什么跨过底。对正方形,面积是 $s(x)^2$(别忘了把整个间隙平方)。切片必须垂直于你沿之积分的那个轴。
A solid has base the region under $y=\sqrt{x}$ from $x=0$ to $4$, with square cross sections perpendicular to the $x$-axis.
- The base of each square is $s(x)=\sqrt{x}$ (curve to $x$-axis).
- Cross-sectional area: $A(x)=(\sqrt{x})^2=x$.
- $V=\displaystyle\int_0^4 x\,dx=\Big[\tfrac{x^2}{2}\Big]_0^4=8$.
一个立体的底是 $y=\sqrt{x}$ 从 $x=0$ 到 $4$ 下方的区域,横截面是垂直于 $x$ 轴的正方形。
- 每个正方形的底是 $s(x)=\sqrt{x}$(曲线到 $x$ 轴)。
- 横截面积:$A(x)=(\sqrt{x})^2=x$。
- $V=\displaystyle\int_0^4 x\,dx=\Big[\tfrac{x^2}{2}\Big]_0^4=8$。
A solid's volume is $V=\int_a^b A(x)\,dx$, the integral of its cross-sectional area. For square cross sections with base $s(x)$, $A(x)=s(x)^2$; for rectangles, $A(x)=s(x)\cdot h(x)$. Find the base $s(x)$ (often the gap between curves), write $A(x)$, and integrate.
立体的体积是 $V=\int_a^b A(x)\,dx$,即其横截面积的积分。对底为 $s(x)$ 的正方形横截面,$A(x)=s(x)^2$;对矩形,$A(x)=s(x)\cdot h(x)$。求出底 $s(x)$(常是曲线间的间隙),写出 $A(x)$,再积分。